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SageMath
E = EllipticCurve("he1")
E.isogeny_class()
Elliptic curves in class 212160he
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212160.f1 | 212160he1 | \([0, -1, 0, -1388481, 632185281]\) | \(-1129285954562528881/4130500608000\) | \(-1082785951383552000\) | \([]\) | \(4976640\) | \(2.3213\) | \(\Gamma_0(N)\)-optimal |
| 212160.f2 | 212160he2 | \([0, -1, 0, 3065919, 3301353921]\) | \(12158099101398341519/25007954601383520\) | \(-6555685251025081466880\) | \([]\) | \(14929920\) | \(2.8706\) |
Rank
sage: E.rank()
The elliptic curves in class 212160he have rank \(1\).
Complex multiplication
The elliptic curves in class 212160he do not have complex multiplication.Modular form 212160.2.a.he
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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
