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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 4025.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4025.e1 | 4025a3 | \([1, -1, 0, -3092, 66941]\) | \(209267191953/55223\) | \(862859375\) | \([2]\) | \(2560\) | \(0.69909\) | |
| 4025.e2 | 4025a2 | \([1, -1, 0, -217, 816]\) | \(72511713/25921\) | \(405015625\) | \([2, 2]\) | \(1280\) | \(0.35251\) | |
| 4025.e3 | 4025a1 | \([1, -1, 0, -92, -309]\) | \(5545233/161\) | \(2515625\) | \([2]\) | \(640\) | \(0.0059379\) | \(\Gamma_0(N)\)-optimal |
| 4025.e4 | 4025a4 | \([1, -1, 0, 658, 5191]\) | \(2014698447/1958887\) | \(-30607609375\) | \([2]\) | \(2560\) | \(0.69909\) |
Rank
sage: E.rank()
The elliptic curves in class 4025.e have rank \(0\).
Complex multiplication
The elliptic curves in class 4025.e do not have complex multiplication.Modular form 4025.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
