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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 55770bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55770.bc7 | 55770bc1 | \([1, 0, 1, -1929984, -987352754]\) | \(164711681450297281/8097103872000\) | \(39083173843304448000\) | \([2]\) | \(2322432\) | \(2.5187\) | \(\Gamma_0(N)\)-optimal |
| 55770.bc6 | 55770bc2 | \([1, 0, 1, -5391104, 3542561102]\) | \(3590017885052913601/954068544000000\) | \(4605106634796096000000\) | \([2, 2]\) | \(4644864\) | \(2.8653\) | |
| 55770.bc3 | 55770bc3 | \([1, 0, 1, -154435584, -738714697394]\) | \(84392862605474684114881/11228954880\) | \(54200020475377920\) | \([2]\) | \(6967296\) | \(3.0680\) | |
| 55770.bc8 | 55770bc4 | \([1, 0, 1, 13590976, 22919468366]\) | \(57519563401957999679/80296734375000000\) | \(-387577000151859375000000\) | \([2]\) | \(9289728\) | \(3.2118\) | |
| 55770.bc5 | 55770bc5 | \([1, 0, 1, -79751104, 274093985102]\) | \(11621808143080380273601/1335706803288000\) | \(6447201619471747992000\) | \([2]\) | \(9289728\) | \(3.2118\) | |
| 55770.bc2 | 55770bc6 | \([1, 0, 1, -154449104, -738578891698]\) | \(84415028961834287121601/30783551683856400\) | \(148586324319603226227600\) | \([2, 2]\) | \(13934592\) | \(3.4146\) | |
| 55770.bc4 | 55770bc7 | \([1, 0, 1, -132171524, -959109111634]\) | \(-52902632853833942200321/51713453577420277500\) | \(-249610963148574392219497500\) | \([2]\) | \(27869184\) | \(3.7611\) | |
| 55770.bc1 | 55770bc8 | \([1, 0, 1, -176943004, -509357053138]\) | \(126929854754212758768001/50235797102795981820\) | \(242478597577949570212612380\) | \([2]\) | \(27869184\) | \(3.7611\) |
Rank
sage: E.rank()
The elliptic curves in class 55770bc have rank \(1\).
Complex multiplication
The elliptic curves in class 55770bc do not have complex multiplication.Modular form 55770.2.a.bc
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
