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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 1035.c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1035.c1 | 1035g3 | \([1, -1, 1, -271202, -54271924]\) | \(3026030815665395929/1364501953125\) | \(994721923828125\) | \([2]\) | \(9600\) | \(1.8356\) | |
| 1035.c2 | 1035g4 | \([1, -1, 1, -149072, 21805544]\) | \(502552788401502649/10024505152875\) | \(7307864256445875\) | \([2]\) | \(9600\) | \(1.8356\) | |
| 1035.c3 | 1035g2 | \([1, -1, 1, -19697, -550456]\) | \(1159246431432649/488076890625\) | \(355808053265625\) | \([2, 2]\) | \(4800\) | \(1.4890\) | |
| 1035.c4 | 1035g1 | \([1, -1, 1, 4108, -64834]\) | \(10519294081031/8500170375\) | \(-6196624203375\) | \([4]\) | \(2400\) | \(1.1424\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1035.c have rank \(0\).
Complex multiplication
The elliptic curves in class 1035.c do not have complex multiplication.Modular form 1035.2.a.c
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
