Table of curves with irrational j

More about isogenies and torsion

The height proxy H and the search

Curves with rational j, and the m=6 table

References

- F is real quadratic field of discriminant
D<10 ;^{6} - The discriminant Δ of E is a unit; and
- The j-invariant is not in
**Q**.

I list each curve with some additional information in the format

- D = disc(F/
**Q**), so D is the fundamental discriminant for whichF = ;**Q**(D^{1/2}) - σ = ±1, and m, are such that
Δ = σ η ;^{m} - s = ±1 is the norm of Δ
(which is +1 unless η has norm −1 and m is odd),
so the Galois conjugate curve has discriminant
ιΔ = s σ η ;^{−m} - H = D
^{−1/2}((c_{4}|Δ|^{−1/3})^{2}+ (ιc_{4}|ιΔ|^{−1/3})^{2}), a proxy for the height of E that measures the difficulty of finding the curve using our search technique (here c_{4}is the usual covariant in12 , and ι is the Galois conjugation of F);^{3}Δ = c_{4}^{3}− c_{6}^{2} - q is a string, usually empty, but taking the value
`Qcurve_`*l*(*l*=2,3,5) if E has an defined over F to a curve E' whose*l*-isogenyj-invariant is the Galois conjugate of j(E); and - the integer d, usually equal 1, is the product of the degrees
of all 2-, 3-, 5-, or
7-isogenies involving E.

The remaining cases are those of the six smallest discriminants
appearing in the table, D=24, 28, 29, 33, 41, and 65.
The curves for D=24, 28, 33 have complex multiplication (CM)
and are 2- or **Q**-curves.

i) For D=24 there are two CM curves each with endomorphism ring
**Z**[3√−2]

ii) For D=28 there are two CM curves, each with endomorphism ring
**Z**[2√−7]**Z**[√−7]^{3}^{3}^{3}**Q**(√−7)

iii) For D=33 there are two CM curves each with endomorphism ring
**Z**[3(1+√−11)/2]^{3}.
[…]

Next the non-CM curves with more than one isogeny:

i) For D=29 there are two isogenous
pairs of **Q**-curves,

ii) For D=41 there is a **Q**-curve**Q**-curves

iii) Finally, each of the D=65 curves is **Q**-curves^{3}^{3}

It is shown in [Comalada 1990, Theorem 2] that if an elliptic curve
has a global minimal model of unit discriminant over a quadratic
number field, and has *all* its ^{3} for D=28 and 17^{3}, 257^{3} for D=65).
[That theorem also determines all curves with full

In several cases the table contains a curve E with an isogeny to
a curve E', but E' does not appear in the table. Usually
this is because E' has discriminant
^{m}**Q**-curve_{F} and unit discriminant.

For a general elliptic curve E over a number field F,
it is known that there is an obstruction δ,
contained in the _{F} that is minimal at each prime.
At best one can choose an ideal D in the class of δ
such that E has good reduction at each prime in D,
and find an equation for E whose discriminant generates
the ideal ^{12}Δ(E)_{i} in D^{i}**Q**-curves*l* of the isogeny splits in F
and the kernel of isogeny reduces to the identity modulo one of the
primes above *l* but not the other. If these primes
are not principal then the isogenous curve has nontrivial δ.

*Torsion.*
Whenever two curves over F are related by isogenies of degree 2,
the kernel of the isogeny contains an

- The
**Q**-curve[w, −2−2*w, 1+w, 3, −3−2*w] forD=29 has rational3-torsion points[1+w, −3−w] and[1+w, −5−2*w] in the kernel of its3-isogeny ; - The
**Q**-curve[w, −2−2*w, 1+w, 3, −3−2*w] forD=41 has rational4-torsion points[1,1] and[1, −2−w] ; - The curve
[1, 0, 1+w, −7−w, 42+6*w] forD=733 has rational3-torsion points[0, 6] and[0, −7−w] in the kernel of its3-isogeny .

This still leaves 14 of the 621 tabulated curves that were beyond
the search limit. Four of them, the CM curves for D=24 and D=28,
come from [Colamada 1990], which suggested the two CM curves with D=33;
two more, the **Q**-curves**Q**-curves

I wouldn't be surprised if there are on the order of 10-20 further curves,
including some **Q**-curves*
[Added a few days later: [Umegaki 1998]
gives such a construction and gives several examples, including
one with a 7-isogeny for D=497. See Table 1
on page 197 for a list of discriminants up to 1000.
I'll add those in due course. Umegaki also reports (p.192, footnote)
that R.Pinch proved in his Ph.D. thesis (Oxford, 1982) that there are
no 13-isogenous *

Almost 90% of the curves with rational j are accounted for by six CM
values of the ^{3}^{3}^{3}^{3}^{3}^{3}^{3}^{3}^{3}^{3}^{3}^{3}

The CM invariants
^{3}^{3}**Q**(7^{1/2})

There are 20 more examples in the m=6 list where more than one curve
appears for the same D. In all but one of these cases, namely
those with D = 316, 5980, 7516, 8284, 41756, 55580, 63964, 71164,
112252, 126076, 167356, 252316, 291004, 374908, 380060, 408380, 413116,
490076, and 672412, there are two curves related by a
^{3}^{3}^{3}^{3}

[Cremona 1992] Cremona, J.E.: Modular symbols for Γ

[Kagawa 1998] Kagawa, T.: Determination of elliptic curves with everywhere good reduction over

[Setzer 1981] Setzer, B.: Elliptic curves with good reduction everywhere over quadratic fields and having rational

[Stroeker 1983] Stroeker, R.J.: Reduction of elliptic curves over imaginary quadratic number fields,

[Umegaki 1998] Umegaki, A.: A Construction of Everywhere Good