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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 75712q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 75712.e2 | 75712q1 | \([0, 1, 0, -225, -71681]\) | \(-4/7\) | \(-2214308282368\) | \([2]\) | \(150528\) | \(1.0478\) | \(\Gamma_0(N)\)-optimal |
| 75712.e1 | 75712q2 | \([0, 1, 0, -27265, -1721121]\) | \(3543122/49\) | \(31000315953152\) | \([2]\) | \(301056\) | \(1.3944\) |
Rank
sage: E.rank()
The elliptic curves in class 75712q have rank \(1\).
Complex multiplication
The elliptic curves in class 75712q do not have complex multiplication.Modular form 75712.2.a.q
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
