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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 344760j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 344760.j6 | 344760j1 | \([0, -1, 0, 186689, -2155828364]\) | \(9317458724864/26001416731875\) | \(-2008061956706637390000\) | \([2]\) | \(16515072\) | \(2.7666\) | \(\Gamma_0(N)\)-optimal |
| 344760.j5 | 344760j2 | \([0, -1, 0, -23947356, -44168373900]\) | \(1229125878116884816/29018422265625\) | \(35856993729924900000000\) | \([2, 2]\) | \(33030144\) | \(3.1131\) | |
| 344760.j4 | 344760j3 | \([0, -1, 0, -53079576, 84328022076]\) | \(3346154465291614084/1315155029296875\) | \(6500354182968750000000000\) | \([2]\) | \(66060288\) | \(3.4597\) | |
| 344760.j2 | 344760j4 | \([0, -1, 0, -380959856, -2861853828900]\) | \(1237089966354690271204/714574355625\) | \(3531892665241549440000\) | \([2, 2]\) | \(66060288\) | \(3.4597\) | |
| 344760.j3 | 344760j5 | \([0, -1, 0, -378762856, -2896495246100]\) | \(-607905111321334101602/14874581985380325\) | \(-147039777174060280080230400\) | \([2]\) | \(132120576\) | \(3.8063\) | |
| 344760.j1 | 344760j6 | \([0, -1, 0, -6095356856, -183164793731700]\) | \(2533559197411478296569602/845325\) | \(8356295307110400\) | \([2]\) | \(132120576\) | \(3.8063\) |
Rank
sage: E.rank()
The elliptic curves in class 344760j have rank \(0\).
Complex multiplication
The elliptic curves in class 344760j do not have complex multiplication.Modular form 344760.2.a.j
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
