# Elliptic Curves in Nature

## Noam D. Elkies

``` =====================================================================
Introduction
Conductors:
11 14 15 21
27 30 32 36
37 43 49 56
77 88 102 112
121 147 162 389
400 446 552 900
1944 2700 4830 5077

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We illustrate the ubiquity of elliptic curves over Q by listing
some of our favorite examples of elliptic curves that arise naturally
in various mathematical (usually Diophantine) problems.

Elliptic curves are identified by Cremona's scheme: the conductor,
followed by a letter that names the isogeny class, followed by a number
that indexes curves within this class.  See

J.E. Cremona: _Algorithms for modular elliptic curves_.
Cambridge University Press, 1992.

for conductors up to 1000, and the online tables beyond that
(e.g. this table covers conductors up to 8000).
For conductors less than 200, we also give in parenthesis
the identifying letter from Tingley's "Antwerp" tables
(Modular Forms in One Variable IV, Springer LNM 476 (1975),
henceforth [MF4]; scanned by William Stein and put online here).
In each case we also give the [a1,a2,a3,a4,a6] vector, the rank,
and the structure of the torsion group (e.g. full rational 2-torsion
is "T=2*2", a 4-torsion point is "T=4").  A curve with CM
(complex multiplication) whose endomorphism ring is generated
by (-D+sqrt(-D))/2 is said to have CM-D; for instance the curves
with CM-3 and CM-4 are those of j-invariant 0 and 1728 respectively.

=================================================================
Isogeny class 11-A [cuspform: the eta product (1,1,11,11)]

11-A1(B): [0,-1,1,-10,-20]  (r=0, T=5)
X_0(11), parametrizes elliptic curves with an 11-isogeny,
or equivalently degree-11 rational functions with four branch points
of type 222221 and whose geometric Galois group is 22-element dihedral.
The 5 rational points are: the two cusps; one point parametrizing
11-isogenies between CM-11 curves, namely

121-B1(D): [0,-1,1,-7,10]  and   121-B2(E): [0,-1,1,-887,-10143]

which are Q(sqrt(-11)) twists of each other; and a pair
parametrizing 11-isogenies between non-CM curves such as

121-C1(F): [1,1,0,-2,-7]  and  121-C2(G): [1,1,0,-3632,82757]

or the Q(sqrt(-11)) twists of these isogenies

121-A1(H): [1,1,1,-30,-76]  and  121-A2(I): [1,1,1,-305,7888].

See the "Remarks on isogenies" preceding Tingley's "Antwerp" tables
[MF4].

This curve, together with its degree-12 Belyi map X_0(11) -> X(1)
[branch points of type (2,2,2,2,2,2), (3,3,3,3), and (11,1)],
recurs as a Shimura curve for the (2,3,11) triangle group,
which comes from a quaternion algebra over the cyclic quintic field
K=Q(cos(2*Pi/11)) ramified at four of the field's five real places
and at no finite prime.  Instead of Gamma_0(11), we use Gamma_0(p)
for the prime p above 11 in K.  The involution w_p is not the same
as for the classical modular curve X_0(11), though: it fixes
the simple pole rather than switching it with the 11-fold pole.

Also: the curve X_0(22)/w_{22}, parametrizing pairs of
22-isogenous Q-curves.  [The curve X_0(22) has genus 2 with model
y^2 = x^6 + 12 x^5 + 56 x^4 + 148 x^3 + 224 x^2 + 192 x + 64
= (x^3 + 4 x^2 + 8 x + 4) (x^3 + 8 x^2 + 16 x + 16)
where x = eta(1,11/2,22)^2; the hyperelliptic involution is w_{11};
the involution w_{22} acts by (x,y) <--> (4/x, 8y/x^3);
for the quotient by w_2, see 11-A3(A) below.  Note that
since the Jacobian J_0(22) is isogenous to the product of these
two elliptic curves, we have a natural example of a genus-2 curve with
bad reduction at 2 whose Jacobian has good reduction at that prime.]
The rational points are: the two cusps; a fixed point of w_2,
which is CM-88; and a pair of points that yield Q-curves
over Q(sqrt(33)), namely those with j = 1025267 + 178618*sqrt(33)
and its conjugate as well as the pair of j-invariants 2- and
11-isogenous with these.

Also: the curve X_1(22)/w_2, parametrizing 2-isogenous pairs of Q-curves
with an 11-torsion point.  The 5-element group  (Z/11Z)* / {1,-1}
acts freely, giving translations by the 5-torsion points.
These points are all cusps (and there also 5 irrational cusps),
so there is no such pair of Q-curves.

11-A3(A): [0,-1,1,0,0]  (r=0, T=5)
X_1(11), parametrizes elliptic curves with an 11-torsion point,
or equivalently degree-11 rational functions with four branch points
as above whose arithmetic Galois group is also 22-element dihedral.
The 5-element group  (Z/11Z)* / {1,-1}  acts freely, giving
translations by the 5-torsion points on X_1(11).  The rational points
are all cusps (there are also 5 irrational cusps), so there is
no elliptic curve over Q with a rational 11-torsion point.

Also: the curve X_0(22)/w_2, parametrizing 11-isogenies
between 2-isogenous pairs of Q-curves.  (See 11-A1(B) above
for the curve X_0(22) and its involutions.)  The rational points are:
the two cusps; a fixed point of w_{11}, which is CM-8;
and a pair of points that yield Q-curves defined over Q(sqrt(-7)),
which turn out to be CM-7 (NB the curves have rational j-invariant,
namely -15^3, but the isogenies cannot be defined over Q).
The degree-12 map X_0(22)/w_2 -> X_0(2)/w_2 is a Belyi map
with branch points of type (2,2,2,2,2,1,1), (4,4,4), and (11,1)],

Also: the curve X_0(55)/w_{55}, parametrizing pairs of 55-isogenous
Q-curves.  The rational points are: the two cusps; a fixed point
of w_{11}, which is CM-11; and a pair of points that yield Q-curves
defined over Q(sqrt(-19)), which turn out to be CM-19 (NB the curves
have rational j-invariant, namely -96^3, but the isogenies cannot be
defined over Q).

11-A2(C): [0,-1,1,-7820,-263580]  (r=0, T=1)
Sometimes called "X_2(11)", a harmless and somewhat humorous
abuse of notation, apparently due to Mazur [reference?].
This is the unique curve of minimal conductor with trivial
Mordell-Weil group.  The curve is also noteworthy for
the massive cancellation in the identity  c_4^3 - c_6^2 = 1728 Delta
-- here  (c_4,c_6,Delta) = (375376, 229985128, -11).  This is likely
the only case of an elliptic curve E/Q whose minimal model has
|a_4| > |Delta|^3 (the uniqueness would follow from a sufficiently
effective ABC conjecture; the ABC threshold has  log|Delta|
asymptotic to  2 log |a_4|).  See

N.D. Elkies:
Rational points near curves and small nonzero |x^3-y^2|
via lattice reduction, Lecture Notes in Computer Science 1838
(proceedings of ANTS-4, 2000; W.Bosma, ed.), 33--63 [arXiv].

for an infinite family of elliptic curves with large cancellation
whose first member is 11-A2(C), built on an observation of Danilov:

L.V. Danilov:
The Diophantine equation  x^3 - y^2 = k  and Hall's conjecture,
Math. Notes Acad. Sci. USSR  32 (1982), 617--618.

The next example is

1342-C3: [1,1,1,-117257250,-488766109679]

It turns out that all curves in this family have rational 25-isogenies
(and conductor divisible by 11); at least the 5-isogeny is there for
good reason, as explained in the  paper cited above.

=================================================================
Isogeny class 14-A [cuspform: the eta product (1,2,7,14)]

14-A1(C): [1,0,1,4,-6]  (r=0, T=6)
X_0(14), parametrizes elliptic curves with a 14-isogeny,
or equivalently degree-14 rational functions with two branch points
of type 2^7 and two of type 2^6 1^2, and whose geometric Galois group
is 28-element dihedral.  The six rational points are the four cusps
and the two points, fixed by the involution w_7, that parametrize
14-isogenies between CM-7 and CM-28 curves (see conductor 49
for the first example).

14-A4(A): [1,0,1,-1,0]  (r=0, T=6)
X_1(14), parametrizes elliptic curves with a 14-torsion point.
The 3-element group  (Z/14Z)* / {1,-1}  acts freely,
giving translations by the 3-torsion points on X_1(14).
The six torsion points on X_1(14) are all cusps
(and there are also six cusps not defined over Q).

14-A2(D): [1,0,1,-36,-70]  (r=0, T=6)
X_0(14)/w_2, parametrizes 7-isogenies between 2-isogenous pairs
of Q-curves.  Three of the rational points lift to pairs
of rational points of X_0(14); these include the two cusps
of X_0(14)/w_2, and one of the fixed points of w_7.
The other three rational points of X_0(14)/w_2
lift to pairs of points defined over Q(sqrt(-7)),
such as the 6-torsion points with x^2+3x+4=0.
Of these three, one is fixed by w_7 and yields CM-7 curves:
the curves, and the degree-7 isogenies, can be defined over Q,
but the 2-isogenies, corresponding to the two ideals above 2
in Q(sqrt(-7)), can be defined only over the CM field.
For the remaining two points, which are switched by w_7,
see the next curve 14-A6(B).

14-A6(B): [1,0,1,-11,12]  (r=0, T=6)
X_1(14)/w_2, parametrizes 2-isogenous pairs of Q-curves
with a 7-torsion point.  Again the 3-element group
(Z/14Z)* / {1,-1}  acts freely, giving translations
by the 3-torsion points on X_1(14).  These 3-torsion points
are the rational cusps (there are also 3 cusps not defined over Q);
the other rational points constitute a single orbit under the action
of  (Z/14Z)* / {1,-1},  and correspond to the curve
E: [2+sqrt(-7), 5+sqrt(-7), 5+sqrt(-7), 0, 0]
and its Galois conjugate, each with a 14-torsion point at [-2,-4].
This curve E is a quotient of the Jacobian of J_0(98) -- indeed so is
its restriction of scalars to Q (which is isogenous to E^2 because
E is isogenous with its Galois conjugate), which corresponds to a pair
of Hecke eigenforms of level 98 with coefficients in Z[sqrt(2)].
(The formula for E is obtained from the generic curve with
a 7-torsion point

[1+d-d^2,d^2-d^3,d^2-d^3,0,0]

by setting d=(3+sqrt(-7))/2.  For this explicit formula for
the universal elliptic curve over X_1(7), see for instance p.195 of

J.Tate, The Arithmetic of Elliptic Curves,
Invent. Math. 23 (1974), 179--206.

For any d, the curve has a 7-torsion point at [0,0]
with a horizontal tangent.)

[For X_1(22)/w_2, see 11-A1(B) above.  The curve X_1(18)/w_2
is rational; the first example of 2-isogenous Q-curves with
an 18-torsion point is  [1, 1+w, 1+w, -101-43*w, 467+197*w]
where w=quadgen(33), with an 18-torsion point at [3+w,0].]

=================================================================
Isogeny class 15-A [cuspform: the eta product (1,3,5,15)]

15-A1(C): [1,1,1,-10,-10]  (r=0, T=4*2)
X_0(15), parametrizes elliptic curves with a 15-isogeny,
or equivalently degree-15 rational functions with four branch points
of type 2^7.1 and whose geometric Galois group is 30-element dihedral.
The Atkin-Lehner group <w_3,w_5> splits the eight rational points
into 2 four-point orbits: the four cusps, and four rational points
that parametrize 15-isogenies between non-CM curves
(see conductor 50 for the first examples).

[conductor 50 and Bring]
[ax^5+bx^2+c with dihedral Galois group, see also 15-A4(F)]
[Frey curve for 9+16=25 !]

15-A8(A): [1,1,1,0,0]  (r=0, T=4)
X_1(15), parametrizes elliptic curves with a 15-torsion point.
The 4-element group  (Z/15Z)* / {1,-1}  acts freely,
giving translations by the 4-torsion points on X_1(15).
The four torsion points on X_1(15) are all cusps.

15-A3(B): [1,1,1,-5,2]   (r=0, T=4*2)
The curve intermediate between X_0(15) and X_1(15);
Frey curve for 1+15=16.

15-A4(F): [1,1,1,35,-28]  (r=0, T=8)
[ax^5+bx^2+c with solvable Galois group; sextic Belyi function
9xy-x^3-15x^2-36x+32 ramified above infty, 0, and 3125/4 (NDE'91).
The quintuple point above 0 is the 8-torsion point (2,6).]

B.K. Spearman, K.S. Williams:
On solvable quintics X^5+aX+b and X^5+aX^2+b,
Rocky Mountain J. Math. 26 #2 (1996), 753--772.

NB the Antwerp tables wrongly give the size of the torsion group
of this curve as 4; presumably this entry was switched with the one
for 15-A2(E): [1,1,1,-135,-660], where Antwerp says |T|=8 but in fact
T=2*2 (Frey curve for 1+80=81).

=================================================================
21-A1(B): [1,0,0,-4,-1]  (r=0, T=4*2)
X_0(21), parametrizes elliptic curves with a 21-isogeny, or equivalently
degree-21 rational functions with four branch points of type 2^10.1
and whose geometric Galois group is 42-element dihedral.
The involution w_7 acts by translation by the 2-torsion point [-2,1];
see 21-A3(C) below for the quotient.  The action of <w_3,w_7>
splits the eight rational points into two orbits: the four cusps,
and four points parametrizing sets of four elliptic curves over Q
related by 3-, 7-, and 21-isogenies.  All such sets have the same
j-invariants, and thus are realted by quadratic twists.
The smallest conductor of curves with a rational 21-isogeny is 162.
See isogeny class 162-B below for another explanation
of these 21-isogenies using the identification of the curves
in that isogeny class with certain Shimura modular curves.

Also: Frey curve for 7+9=16, and 2-isogenous with the curve
21-A2(D): [1,0,0,-49,-136] (r=0, T=2*2) which is Frey for 1+48=49.

21-A3(C): [1,0,0,-39,90] (r=0, T=8)
X_0(21)/w_7, parametrizes 3-isogenies between 7-isogenous pairs
of Q-curves.  Four of the rational points lift to pairs
of rational points of X_0(21).  The four new rational points
lift to pairs of points defined over Q(sqrt(-3)), such as
the 8-torsion points with x^2+5x+13=0 and y=8.  Two of the four
new rational points are fixed by w_3; one of these is a CM-3 point,
the other CM-12.  The remaining two points are switched by w_3,
and are also CM, giving 3-isogenies between CM-3 and CM-27 curves.

A-B-D-E
| |
C F

21-A1(B) [1,0,0,-4,-1]	8
21-A2(D) [1,0,0,-49,-136]4 [Frey for 1+48=49]
21-A3(C) [1,0,0,-39,90]	8
21-A4(A) [1,0,0,1,0]	4
21-A5(F) [1,0,0,-784,-8515]2
21-A6(E) [1,0,0,-34,-217]2

=================================================================
Isogeny class 27-A [cuspform: the eta product (3,3,9,9)]

27-A1(B): [0,0,1,0,0]  (r=0, T=3)
27-A2(D): [0,0,1,-270,-1708]  (r=0, T=1)
27-A3(A): [0,0,1,0,-7]  (r=0, T=3)
27-A4(C): [0,0,1,-30,63]  (r=0, T=3)

The CM curves of minimal conductor.  The curves 27-A1(B) and 27-A3(A)
have CM-3, are each other's quadratic twist by Q(sqrt(-3)),
and are 3-isogenous to each other.  The curves 27-A2(D) and 27-A4(C)
have CM-27, are each other's quadratic twist by Q(sqrt(-3)),
and are 27-isogenous to each other.

The "strong Weil curve" 27-A1(B) is X_0(27) itself: it parametrizes
elliptic curves with a cyclic 27-isogeny, or equivalently
degree-27 rational functions with four branch points of type 2^13.1
and whose geometric Galois group is 54-element dihedral.
The complex multiplication by a cube root of unity arises from
the map z |--> z+1/3 of the upper half-plane, which normalizes
Gamma_0(27).  The three rational points include two of the six cusps.
This there is a unique non-cusp rational point, and a unique cyclic
27-isogeny between elliptic curves over Q up to quadratic twist.
The minimal conductor of such an isogeny is 27, for the isogeny
between 27-A2(D) and 27-A4(C) noted above.

The curve 27-A3(A) is isomorphic with the Fermat cubic  X^3+Y^3=Z^3.
Euler proved that the curves 27-A3(A) and 27-A1(B) have no rational
points other than their torsion points, using what we now recognize
as descent via the 3-isogeny between these curves.  He thus proved
"Fermat's last theorem" for exponent 3.  This was the first example
of a 3-descent.

The isomorphism with  X^3+Y^3=Z^3  shows that 27-A3(A) admits
a Belyi function of degree 3, for instance Y/X.  This is
the minimal degree of a Belyi function on an elliptic curve,
and is attained only by this curve and its cubic twists
E_d: X^3+Y^3=d*Z^3.  When E_d has positive rank, this yields
an infinite family of ABC examples  (f-16 : 32 : f+16)
with ABC ratio at least 1-o(1).  This first happens for d=6,
d=7, and d=9.  [This construction, but not the connection
with the Belyi map, appears in

L. Szpiro: Discriminant et conducteur des courbes elliptiques,
Ast\'erisque 183 (1990), 7--18.

for the Belyi connection, see the reference given under curve
400-H1 below.]

=================================================================
30-A2(B): [1,0,1,-19,26]  (r=0, T=6*2)
parametrizes degree-5 rational functions with Galois group A5
and four branch points of type 311, all rational.
[NDE 2004, unpublished]

30-A6(F): [1,0,1,-334,-2368]  (r=0, T=2*2)
Frey curve for 3+125=128.

N.D. Elkies: Wiles minus epsilon implies Fermat, pages 38--40 in
_Elliptic Curves, Modular forms, and Fermat's Last Theorem_
(J.Coates and S.T.Yau, eds.; Boston: International Press, 1995;
Proceedings of the 12/93 conference on elliptic curves
and modular forms at the Chinese University of Hong Kong).

The curve 30-A2(B) is likewise the Frey curve for 5+27=32.

=================================================================
Isogeny class 32-A [cuspform: the eta product (4,4,8,8)]

32-A1(B): [0,0,0,4,0]  (r=0, T=4)
32-A2(A): [0,0,0,-1,0]  (r=0, T=2*2)

These are the curves of minimal conductor with CM-4,
and are 2-isogenous to each other.  (There are
two further curves in the isogeny class, namely
[0,0,0,-11,14] and [0,0,0,-11,-14], both with CM-16.)
Each curve is isomorphic with its own quadratic twist
by the CM field Q(sqrt(-1)).

The "strong Weil curve" 32-A1(B) is X_0(32) itself.
The complex multiplication by sqrt(-1) arises from
the map z |--> z+1/4  of the upper half-plane, which
normalizes Gamma_0(32).  The four rational points are all cusps
(and there are also four cusps not defined over Q).  This curve
is also isomorphic with the curve y^2=x^4-1, the quotient of
the Fermat quartic by an involution.

The curve 32-A2(A) parametrizes rational Pythagorean triangles
of area 1, and three-term arithmetic progressions of rational squares
whose congruum (common difference) is 1.  More generally, for
nonzero d the quadratic twist [0,0,0,-d^2,0] by Q(sqrt(-d))
parametrizes rational Pythagorean triangles of area d,
and three-term arithmetic progressions of rational squares
whose congruum is d.  Such triangles and progressions exist
if and only if the twisted curve has positive rank, in which case
d is said to be "congruent" (this use of the word long predates
Gauss' invention of modular notation).

Fermat proved that these curves have no rational points other than
their torsion points, using what he famously called his "method of descent",
and which is a special case of what we now call a descent via a 2-isogeny,
here the 2-isogeny between 32-A1(B) and 32-A2(A).  Fermat thus proved
that 1 is not a congruent number, settling an open question going back
several centuries to Fibonacci -- and also proved the case n=4 of what
we now call Fermat's Last Theorem.

More recently (but still two generations ago), in his paper

K. Heegner: Diophantische Analysis und Modulfunktionen.
Math. Z. 56 (1952), 227--253.

Heegner used the isogeny with X_0(32) to construct rational points
on certain quadratic twists of 32-A2(A), and thus prove certain
numbers are ``congruent'', starting from algebraic points on X_0(32)
that parametrize 32-isogenies between suitable CM curves.  This
was the first example of such a use of CM points (now often called
``Heegner points'' in this context) to study the arithmetic
of elliptic curves (and, more recently, also the arithmetic of
other abelian varieties occurring in the Jacobians of modular curves).

=================================================================
Isogeny class 36-A [cuspform: the eta product (6,6,6,6)]

36-A1(A): [0,0,0,0,1]  (r=0, T=6)
36-A3(C): [0,0,0,0,-27]  (r=0, T=2)

These are the curves of third-smallest conductor with CM-3;
they are each other's quadratic twist by Q(sqrt(-3)), and are
3-isogenous to each other.  They are also the only CM-3 curves
with a rational 2-torsion point; the 2-isogenous curves
[0,0,0,-15,22] and [0,0,0,-135,-594] have CM-12.

The "strong Weil curve" 36-A1(A) is X_0(36) itself.
The complex multiplication by a cube (sixth) root of unity
arises from the map z |--> z+1/3 (resp. z |--> z+1/6)
of the upper half-plane, which normalizes Gamma_0(36).
Conjugation by the map  z |--> 6z  takes Gamma_0(36) to Gamma(6)
and thus identifies this curve with X(6).  The six rational points
are all cusps (and there are also six cusps not defined over Q).

The curve 36-A3(C) is isomorphic with x^3+y^3=2.  In the paper

P. Satge': Un analogue du calcul de Heegner. (French)
[An analogue of Heegner's computation],
Invent. Math. 87 (1987), no. 2, 425--439.

Satge' used the isogeny with X_0(36) to construct
nontrivial rational points on the elliptic curves x^3+y^3=k with
k=2p (p congruent to 2 mod 9) and k=2p^2 (p congruent to 5 mod 9).

=================================================================
37-A1(A): [0,0,1,-1,0]  (r=1, T=1)
The elliptic curve of minimal conductor with positive rank.

Also: the curve X_0(37)/w, parametrizing 37-isogenous pairs
of Q-curves.  There is one rational point, other than the cusp,
that lifts to a pair of rational points on the genus-2 curve X_0(37),
and parametrizes rational 37-isogenies between elliptic curves over Q
such as 1225-H1: [1,1,1,-8,6] and 1225-H2: [1,1,1,-208083,-36621194].
See

J. V\'elu: Les points rationnels de X_0(37),
pages 169-179 in the Proceedings of Journees Arithmetiques 1973
(Grenoble), Bull. Soc. Math. France Mem. 37, 1974.

B. Mazur, H.P.F. Swinnerton-Dyer: Arithmetic of Weil Curves,
Inv. Math. 25 (1974), 1--61.

for this pair of rational points on X_0(37), and the first example
in the Appensix of my paper

N.D. Elkies: Elliptic and modular curves over finite fields
and related computational issues, pages 21--76 in
_Computational Perspectives on Number Theory:
Proceedings of a Conference in Honor of A.O.L. Atkin_
(D.A. Buell and J.T. Teitelbaum, eds.; AMS/International Press, 1998).

for the computation of the isogeny.  The "Remarks on isogenies"
in [MF4] attribute to Velu the computation of the example
[1,1,1,-8,6] of a curve with a 37-isogeny over Q corresponding
to a non-cusp rational point on X_0(37).

Also: The integral points on this curve y^2+y=x^3-x correspond
to solutions of the classical problem of finding all integers
that are simultaneously the product of two consecutive integers
and the product of three consecutive integers.  The fact that
210 = 5*6*7 = 14*15 is the last such example can be proved easily
from the fact that (0,0) generates the group of rational solutions.
See J.H.Silverman, _The Arithmetic of Elliptic Curves_ (Springer
GTM 106, 1985), page 275, exercise 9.13 .

Also: This elliptic curve parametrizes curves C of genus 2
with a rational point P such that 5P, 13P, 45P are also on the curve
(i.e. there are points that differ from those multiples by divisors
equivalent to a multiple of the canonical divisor).  Cf. conductors
77 and 446.  [NDE 2001, unpublished.]

Also: This is the elliptic curve involved in the "Somos-4" sequence
..., 59, 23, 7, 3, 2, 1, 1, 1, 1, 2, 3, 7, 23, 59, ...
satisfying the recursion  S4[n+4] S4[n] = S4[n+1] S4[n+3] + S4[n+2]^2
with S4[n]=1 for n=0,1,2,3.  Let P be the generator (0,0) of
the Mordell-Weil group.  Then x((2n-3)P) has numerator S4[n]^2.
[I don't know the original source of this formula for S4[n],
which can also be expressed in terms of the sigma function
for the lattice of the curve.  It must have been known
no later than 1995; cf. 102-A1(E) below.]

=================================================================
43-A1(A): [0,1,1,0,0]  (r=1, T=1)
The elliptic curve of second-smallest conductor with positive rank.

Also: the curve X_0(43)/w, parametrizing 43-isogenous pairs
of Q-curves.  There is one rational point, other than the cusp,
that lifts to a rational point on the genus-3 curve X_0(43),
namely the fixed point of w that parametrizes a 43-isogeny
between a CM-43 curve and its quadratic twist by Q(sqrt(-43))
(the smallest example is 1849-A1: [0,0,1,-860,9707]
and 1849-A2: [0,0,1,-1590140,-771794326]).

=================================================================
49-A1(A): [1,-1,0,-2,-1]  (r=0, T=2)
49-A2(B): [1,-1,0,-37,-78]  (r=0, T=2)
49-A3(C): [1,-1,0,-107,552]  (r=0, T=2)
49-A4(D): [1,-1,0,-1822,30393]  (r=0, T=2)

The curves 49-A1(A), 49-A3(C) are the elliptic curves of minimal
conductor with CM-7, are each other's twist by Q(sqrt(-7)),
and are 7-isogenous to each other.  The curves 49-A2(B), 49-A4(D)
are the elliptic curves of minimal conductor with CM-28,
are each other's twist by Q(sqrt(-7)), and are 7-isogenous
with each other and 2-isogenous with 49-A1(A), 49-A3(C) respectively.
The 14-isogenies between 49-A1(A) and 49-A4(D), and between
49-A2(B) and 49-A3(C), are the only ones over Q up to quadratic twist;
see 14-A1(C) above.

The "strong Weil curve" 49-A1(A) is X_0(49) itself: it parametrizes
parametrizes elliptic curves with a cyclic 49-isogeny, or equivalently
degree-49 rational functions with four branch points of type 2^24.1
and whose geometric Galois group is 98-element dihedral.
There are none such over Q, because both rational points
are cusps (there are also six irrational cusps).
This is the largest value of N for which X_0(N) has genus 1.

This curve is also a quotient of the 7th Fermat curve in two different
ways.  Either way yields an elementary proof of Fermat's
``Last Theorem'' for exponent 7, using descent via the 2-isogeny
between this curve and 49-A2(B).  One of these proofs was first
obtained by Genocchi in the mid-19th century:

A. Genocchi: Intorno all'equazione  x^7 + y^7 + z^7 = 0.  (Italian)
[concerning the equation  x^7 + y^7 + z^7 = 0.]
Annali di Matematica Pura ed Applicata 6 (1864), 287--288.

He used the identification of the elliptic curve 49-A1(A)
with the quotient of  x^7 + y^7 + z^7 = 0  by the symmetric group
S_3 permuting the coordinates.  The curve 49-A1(A) is also
the quotient of the Klein quartic  X^3 Y + Y^3 Z + Z^3 X = 0
by the 3-element group of cyclic permutations of the coordinates,
as can be seen either directly or by the identification of
the Klein quartic with the modular curve X(7).  Since
the Klein quartic is in turn the image of the 7th Fermat curve
under the map taking (x:y:z) to (xy^3:yz^3:zx^3), this also
yields another proof of the case n=7 of Fermat.  See

N.D. Elkies: The Klein quartic in number theory, pages 51-102 in
_The Eightfold Way: The Beauty of Klein's Quartic Curve_
(S.Levy, ed.; Cambridge Univ. Press, 1999; also
online at the MSRI Publications site).

=================================================================
50-
(15-isogeny; Bring and a Shimura curve)

=================================================================
56-A1(C): [0,0,0,1,2]  (r=0, T=4)
112-B1(A): [0,0,0,1,-2]  (r=0, T=2)

These curves are quadratic twists by Q(sqrt(-1)) and can be found on
the Fermat quartic surface  x^4 + y^4 + z^4 = t^4  (so there are
rational functions x,y,z,t on each curve satisfying that identity).
These are the first curves that occur in Demjanenko's fibration
of the Fermat quartic surface by genus-1 curves.  Their 6 rational
points yield only the trivial points on the surface, such as
(1:0:0:1) and (-1:0:0:1).  See

V.A. Demjanenko: L. Euler's conjecture (Russian),
Acta Arith. 25 (1973--4), 127--135.

and also

N.D. Elkies: On A^4 + B^4 + C^4 = D^4,
Math. of Comp. 51 (Oct. 1988), 825--835.

where another curve in this fibration was used to find the first
nontrivial rational points known on that surface.

=================================================================
77-A1(F) : [0,0,1,2,0]  (r=1, T=1)
Parametrizes curves C of genus 2 with a rational point P
such that 5P, 13P, 53P are also on the curve (i.e. there are points
that differ from those multiples by divisors equivalent to a multiple
of the canonical divisor).  Cf. conductors 37 and 446.
[NDE 2001, unpublished.]

=================================================================
88-A1(A) : [0,0,0,-4,4]  (r=1, T=1)
Parametrizes triangles ABC with rational sides a,b,c for which
the altitude from A, angle bisector to B, and median from C
are concurrent.  Equivalently,  c (a^2 + b^2 - c^2) = a (a^2 + c^2 - b^2)
[proof by standard triangle geometry, including "Ceva's theorem"].
This elliptic curve is put in standard Weierstrass form by taking
c = ((2/x) - 1) a,  when  y^2 = x^3 - 4x + 4.  The M-W generator
(x,y)=(2,2) corresponds to an equilateral triangle.  The higher
multiples that yield positive (a:b:c) are the 7th, 10th, and 12th,
with x-coordinates  10/9, 88/49, 206/961,  and triangles (a:b:c) =
(a:b:c) = (15:13:12),  (308:277:35),  (3193:26447:26598).

I originally listed this as [NDE 2001, unpublished], but evidently
it was known at least as early as 1995, being the title curve of
the article

Richard K. Guy: My Favorite Elliptic Curve: A Tale of Two Types of Triangles,
Amer. Math. Monthly 102 #9 (Nov. 1995), 771--781

which remarkably gives two further natural appearances of this curve in
Diophantine equations arising from plane geometry, both related with
``Heron triangles'' (triangles with rational sides and area):

@ The curve parametrizes isosceles Heron triangles; and

@ (somewhat less decisively) The Heron triangles whose perimeter and area
coincide with the perimeter and area of a rational rectangle are parametrized
by a pencil of elliptic curves (that is, a surface with a rational function t
such that the preimage of the generic t is an elliptic curve; equivalently,
an elliptic curve over Q(t)); the typical curve in this family has
torsion group (Z/2Z) x (Z/2Z), but those that also have a 4-torsion
point are parametrized by the elliptic curve 88-A1.
[R.K.Guy describes this last item as joint work with Andrew Bremner.]

=================================================================
102-A1(E): [1,1,0,-2,0]  (r=1, T=2)
This is the elliptic curve involved in the "Somos-5" sequence
..., 37, 11, 5, 3, 2, 1, 1, 1, 1, 1, 2, 3, 5, 11, 37, ...
satisfying the recursion  S5[n+5] S5[n] = S5[n+1] S5[n+4] + S5[n+2] S5[n+3]
with S5[n]=1 for |n|<3.  Let P be the point (2,2) of infinite order.
Then x(nP) has numerator S5[n]^2 for n even and 2*S5[n]^2 for n odd
(taking x(nP)=1/0 for n=0).

[NDE 1995, publicized on Usenet and preserved by Dave Rusin here
but not formally published.  Also see Don Zagier's
"Problems posed at the St Andrews Colloquium, 1996",
Day 5, Problem 1: solution here).  I don't know if this
was based on my 1995 postings or found independently.  Also cf.
37-A1(A) above for the Somos-4 sequence.]

=================================================================
112-B1(A): [0,0,0,1,-2]  (r=0, T=2)

See conductor 56 for the quadratic twist of this curve by Q(sqrt(-1)).
Both curves occur on the Fermat quartic surface  x^4+y^4+z^4=t^4,
but yield only trivial rational points on the surface.

=================================================================
Conductor 121: see the N=11 curves above.  The preface in [MF4]
to Tingley's "Antwerp" tables attributes to Velu the discovery
of several curves of conductor 11 and 121 too large to be found
by direct search (see p.76).  Each of the curves of conductor 121
is either involved in an 11-isogeny, and so parametrized by X_0(11),
or is itself a Q(sqrt(-11)) twist of one of the curves of conductor 11.
These twists are

121-D1(A): [0,-1,1,-40,-221]
121-D2(B): [0,-1,1,-1250,31239]
121-D3(C): [0,-1,1,-946260,354609639]

For instance, 121-D2(B) parametrizes pairs of Q-curves
that are 11-isogenous and conjugate over Q(sqrt(-11)).
Unfortunately there aren't any such because the curve
has trivial Mordell-Weil group.

According to E. Halberstadt (Sur la courbe modulaire X_{nd\'ep}(11) =
On the modular curve X_{nonsplit}(11), Experimental Math. 7 (1998) #2,
163--174), the CM curve  121-B1(D): [0,-1,1,-7,10]  is also the
modular curvre X_{nonsplit}(11), classifying elliptic curves E
such that the Galois action on E[11] factors through the normalizer
of a nonsplit Cartan subgroup of GL_2(F_{11}).

=================================================================
147-B1(I): [0,1,1,-114,473]  (r=0, T=1)
147-B2(J): [0,1,1,-44704,-3655907]  (r=0, T=1)

These are Shimura modular curves for the (2,3,7) triangle group,
which comes from a quaternion algebra over the cubic field
K=Q(cos(2*Pi/7)) ramified at two of the field's three real places
and at no finite prime.  These curves are respectively the "X_1"
and "X_0" curves for the prime above 3 (more precisely, 147-B1(I)
is the Jacobian of "X_1" but not known to be isomorphic with X_1
over Q).  They are 13-isogenous for much the same reason that
X_0(11) and X_1(11) are 5-isogenous -- NB the prime above 3 has
q=27, and 13=(q-1)/2.  As was true for the elliptic modular curves
of level 11, we see (mutatis mutandis) that the X_1 curve has
a 13-torsion point, but here it's rational over K, not Q.
Over K the curves attain their potentially good reduction at 7,
and thus have conductor 3 as expected.  The X_0 curve
carries a degree-28 rational function that is ramified
only above three points (type 2^14, 3^9.1, 7^4) and realizes
the modular cover X_0(3) -> X(1); this map has been computed explicitly
(which is how the curve 147-B2(J) was first identified with this
Shimura modular curve).  While 13-isogenies are parametrized
by the rational curve X_0(13), they are still quite rare:
these curves and their quadratic twists are the only examples
of conductor up to 1000 according to Cremona's tables.
(In November 2003 William Stein searched the extended Cremona tables
and found that the first examples not obtained from 147-B1/B2 by
quadratic twists have conductor 2450, such as 2450-O1: [1,-1,0,-2,6]
and 2450-O2: [1,-1,0,-6827,-215419].   Also, several other examples,
all of conductor  20736 = 2^8 3^4  and related by quadratic twists,
appeared in Table 4 of [MF4], the list of elliptic curves with
good reduction away from 2 and 3, based on Coghlan's tables.

See this paper, and cf. conductor 162 below.

=================================================================
162-B1(G): [1,-1,1,-5,5]  (r=0,T=3)
162-B3(I): [1,-1,1,-95,-697]  (r=0,T=3)

These are Shimura modular curves for the (2,3,9) triangle group,
which comes from a quaternion algebra over the cubic field
K=Q(cos(2*Pi/9)) ramified at two of the field's three real places
and at no finite prime.  These curves are respectively the "X_1"
and "X_0" curves for the prime above 2 (more precisely, 162-B1(G)
is the Jacobian of "X_1" but not known to be isomorphic with X_1
over Q).  They are 7-isogenous for much the same reason that
X_0(11) and X_1(11) are 5-isogenous -- NB the prime above 2 has q=8,
and 7=q-1.  As was true for the elliptic modular curves
of level 11, we see (mutatis mutandis) that the X_1 curve has
a 7-torsion point, but here it's rational over K, not Q.
Over K the curves attain their potentially good reduction at 3,
and thus have conductor 2 as expected.  The X_0 curve
carries a degree-9 rational function that is ramified
only above three points (type 2^4 1, 3^3, 9) and realizes
the modular cover X_0(2) -> X(1).  Using the 3^3 and 9 points
we see that the curve also has a 3-torsion point.  It is known
that these two elliptic curves are the only elliptic curves over Q
with a rational 3-torsion point and a rational 7-isogeny
(see conductor 21 above).  The modular cover X_0(2) -> X(1)
has been computed explicitly, which is how the curve 162-B1(I)
was first identified with this Shimura modular curve.
Using the model  y^2 + 15 x y + 128 y = x^3  for this curve,
which puts a 3-torsion point at the origin, we have the formula
f = (y-x^2-17x)^3 / 2^14 y  for a degree-9 map branched above 0,1,inf.
See this paper, and cf. conductor 147 above.

=================================================================
389-A1: [0,1,1,-2,0]  (r=2, T=1)
The elliptic curve of minimal conductor with rank >1.

=================================================================
400-H1: [0,0,0,5,10]  (r=1, T=1)
This curve answers in at least two apparently unrelated questions
involving unusual ramification structure of maps between curves.

In

B. Mazur, H.P.F. Swinnerton-Dyer: Arithmetic of Weil Curves,
Inv. Math. 25 (1974), 1--61.

The authors report (p.17) on numerical computations that suggest
that the modular parametrization of this curve has a triple point
at i/20; this might be still the only known case of a modular
parametrization with a branch point of multiplicity higher than 2
(NB generic maps between curves have only double points).

In

N.D. Elkies: ABC implies Mordell,
International Math. Research Notices 1991 #7, 99-109
[bound with Duke Math. J. 64 (1991)].

we report that this curve admits a Belyi map (a rational function
branched above only three points of P^1) of degree 5, namely f=(x-5)*y
ramified above infty, 16, and -16 (with multiplicities (5), (4,1), (4,1)).
Since the curve has positive rank, the values of this function on E(Q)
yield an infinite family of ABC examples  (f-16 : 32 : f+16)
with ABC ratio at least 1-o(1).

=================================================================
446-D1: [1,-1,0,-4,4]  (r=2, T=1)
The elliptic curve of second-smallest conductor with rank >1.

Also: parametrizes curves C of genus 2 with a rational point P
such that 5P, 13P, 51P are also on the curve (i.e., there are points
that differ from those multiples by divisors equivalent to a multiple
of the canonical divisor).  Cf. conductors 37 and 77.
[NDE 2001, unpublishede]

=================================================================
552-E1: [0,1,0,-4,32]  (r=1, T=1)
This curve appears on the surface

x1^2+x2^2+x3^2 = y1^2+y2^2+y3^2
x1^3+x2^3+x3^3 = y1^3+y2^3+y3^3
x1^4+x2^4+x3^4 = y1^4+y2^4+y3^4

in P^5; since the curve has positive rank, it follows
that the surface, though of general type, has infinitely many
rational points.  There are thus infinitely many primitive
pairs of triples of integers (x1,x2,x3) and (y1,y2,y3)
with equal sums of squares, cubes, and fourth powers.
Thanks to Jordan Ellenberg for re-locating the reference:

Ajai Choudhry: Triads of integers with equal sums of squares,
cubes, and fourth powers, Bull.London.Math.Soc. 35 (2003), 821--824.

=================================================================
900-D1: [0,0,0,-120,740]  (r=1, T=1)
Admits a degree-5 Belyi map, namely f=(x+5)y ramified above
infty,162,-162 with multiplicities (5), (3,2), (3,2).
(The divisors of f=162 and f=-162 are 2(3P)+3(-2P), 2(-3P)+3(2P)
respectively where P=(16,54) is a M-W generator).  Can be used
to obtain an infinite ABC family, as with 400-H1
[NDE 1991, unpublished].

=================================================================
1944-H1: [0,0,0,-27,-42]  (r=1,T=1)
1944-C1: [0,0,0,-243,1134]  (r=0,T=1)
The elliptic curves of smallest conductor whose 3-adic
Galois representation is surjective mod 3 but not mod 9.
This property is preserved under quadratic twist;
the two curves are each other's twists by Q(sqrt(-3)),
and the next examples are their twists 3888-A1: [0,0,0,-27,42]
and 3888-B1: [0,0,0,-243,-1134] by Q(sqrt(-1)), both with r=T=1.

Serre showed in 1968 that for p>3 surjectivity mod p implies
surjectivity to the p-adic group GL_2(Z_p), for purely
group-theoretic reasons, and noted that the group-theoretic
argument fails for p=2 and p=3; see Ch.IV, section 3.4,
Lemma 3 and Exercise 3 in

J.-P. Serre:
_Abelian \$l\$-adic Representations and Elliptic Curves._
Wellesley, Mass.: A.K. Peters, 1997.

But it seems the question of actually describing curves for which the
implication (surjective mod p ==> surjective p-adically) fails at p=3
has not been pursued further.  It turns out that such curves are
parametrized by a modular curve of genus 0 whose degree-27 map to X(1)
is represented by the rational function

f(x) = - 3^7 (x^2-1)^3 (x^6+3x^5+6x^4+x^3-3x^2+12x+16)^3
(2x^3+3x^2-3x-5) / (x^3-3x-1)^9.

An elliptic curve over Q has the desired property if and only if
its j-invariant is the value of that function at some point
x in P^1(Q) other than the two zeros x=1, x=-1 of the function.
The curves of conductor 1944 and 3888 have x=infinity and j=4374.
The next examples are 6075-L1: [0,0,1,-135,-604] (r=T=1)
and its quadratic twists of discriminant -3, 5, -15, with x=0,
and 7776-B1: [0,0,0,-5427,-153882] (r=T=1) and its quadratic
twists of discriminant -1, -3, 3, with x=-1/2.  Also x=-2
yields a curve with integral j-invariant 419904 = 2^6 3^8
(the second-smallest nonzero case) but conductor no smaller than
62208 = 2^8 3^5.  This is still too large for Cremona, but
(like the curves of conductor 1944 and 7776) already appeared
in Table 4 of [MF4], the list of elliptic curves with good reduction
away from 2 and 3, based on Coghlan's tables.

[NDE 2004, unpublished but now arXived; thanks to William Stein and
Grigor Grigorov for bringing the Serre results to my attention.]

=================================================================
2700-N1: [0,0,0,-15,-10]  (r=1, T=1)
Admits a degree-6 Belyi map, namely

f = ((3x^2+12x+5)y-10x^3-30x^2-6x+6) / (9x+26)

ramified above 0,inf,-16 with multiplicities (5,1), (5,1), (3,3).
The quintuple zero (-1,-2) generates the Mordell-Weil group.
Can be used to obtain an infinite ABC family, as with 400-H1
[NDE 1991, unpublished].

=================================================================
4830-KK1: [1,0,0,-480,230400]  (r=1, T=6)
Parametrizes a family of curves of genus 4 with at least 126
rational points.  Let  f(x) = (x^3-9x) / (x^2-1)  and
g(z) = f(4z-3)/f(z),  a rational function of degree 3;
then 4830-KK1 parametrizes solutions of g(z)=g(z')=g(z'')=t
in distinct rationals z,z',z''.  For such t, the curve
C_t: f(x) = t f(y)  has genus 4 and 126 generically distinct points.
Since the elliptic curve has positive rank, there are
infinitely many such curves C_t.

[NDE, unpublished, 1998 or earlier.]

=================================================================
5077-A1: [0,0,1,-7,6]  (r=3, T=1)
The elliptic curve of minimal conductor with rank at least 3.
This is the last case for which the minimal conductor has been
rigorously proved, since the only approach we know for proving
such a thing is to calculate one-dimensional factors of J_0(N),
and that project has not yet covered all N<234446 (the smallest
known conductor of an elliptic curve over Q of rank 4).  See

W. Stein, A. Jorza, and J. Balakrishnan:
The Smallest Conductor for an Elliptic Curve of Rank Four is Composite

for the computations that established that there is no
rank-4 curve of prime conductor less than 234446.  See also

J. Buhler, B. Gross, and D. Zagier:  On the conjecture
of Birch and Swinnerton-Dyer for an elliptic curve of rank 3,
Math. Comp. 44 (1985) no. 170, 473--481.

This curve is used to prove the best effective lower bound known
on the class number of an imaginary quadratic field.
```