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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 12635.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12635.e1 | 12635a3 | \([0, -1, 1, -47411, 4172421]\) | \(-250523582464/13671875\) | \(-643205404296875\) | \([]\) | \(42768\) | \(1.5997\) | |
| 12635.e2 | 12635a1 | \([0, -1, 1, -481, -4349]\) | \(-262144/35\) | \(-1646605835\) | \([]\) | \(4752\) | \(0.50107\) | \(\Gamma_0(N)\)-optimal |
| 12635.e3 | 12635a2 | \([0, -1, 1, 3129, 10452]\) | \(71991296/42875\) | \(-2017092147875\) | \([]\) | \(14256\) | \(1.0504\) |
Rank
sage: E.rank()
The elliptic curves in class 12635.e have rank \(1\).
Complex multiplication
The elliptic curves in class 12635.e do not have complex multiplication.Modular form 12635.2.a.e
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.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
