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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 397800.cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 397800.cd1 | 397800cd5 | \([0, 0, 0, -8115120075, 281378259499750]\) | \(2533559197411478296569602/845325\) | \(19719741600000000\) | \([2]\) | \(150994944\) | \(3.8778\) | |
| 397800.cd2 | 397800cd3 | \([0, 0, 0, -507195075, 4396534024750]\) | \(1237089966354690271204/714574355625\) | \(8334795284010000000000\) | \([2, 2]\) | \(75497472\) | \(3.5313\) | |
| 397800.cd3 | 397800cd6 | \([0, 0, 0, -504270075, 4449748549750]\) | \(-607905111321334101602/14874581985380325\) | \(-346994248554952221600000000\) | \([2]\) | \(150994944\) | \(3.8778\) | |
| 397800.cd4 | 397800cd4 | \([0, 0, 0, -70668075, -129516322250]\) | \(3346154465291614084/1315155029296875\) | \(15339968261718750000000000\) | \([2]\) | \(75497472\) | \(3.5313\) | |
| 397800.cd5 | 397800cd2 | \([0, 0, 0, -31882575, 67863087250]\) | \(1229125878116884816/29018422265625\) | \(84617719326562500000000\) | \([2, 2]\) | \(37748736\) | \(3.1847\) | |
| 397800.cd6 | 397800cd1 | \([0, 0, 0, 248550, 3311657125]\) | \(9317458724864/26001416731875\) | \(-4738758199384218750000\) | \([2]\) | \(18874368\) | \(2.8381\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 397800.cd have rank \(0\).
Complex multiplication
The elliptic curves in class 397800.cd do not have complex multiplication.Modular form 397800.2.a.cd
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
