Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 2025.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2025.e1 | 2025f2 | \([1, -1, 0, -50667, 4421366]\) | \(-15590912409/78125\) | \(-72081298828125\) | \([]\) | \(6048\) | \(1.5064\) | |
| 2025.e2 | 2025f1 | \([1, -1, 0, -42, -3259]\) | \(-9/5\) | \(-4613203125\) | \([]\) | \(864\) | \(0.53343\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2025.e have rank \(0\).
Complex multiplication
The elliptic curves in class 2025.e do not have complex multiplication.Modular form 2025.2.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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
