Show commands:
SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 75712bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 75712.cj1 | 75712bp1 | \([0, 1, 0, -1057, -14273]\) | \(-226981/14\) | \(-8063025152\) | \([]\) | \(46080\) | \(0.65530\) | \(\Gamma_0(N)\)-optimal |
| 75712.cj2 | 75712bp2 | \([0, 1, 0, 3103, 845183]\) | \(5735339/537824\) | \(-309749174239232\) | \([]\) | \(230400\) | \(1.4600\) |
Rank
sage: E.rank()
The elliptic curves in class 75712bp have rank \(1\).
Complex multiplication
The elliptic curves in class 75712bp do not have complex multiplication.Modular form 75712.2.a.bp
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
