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SageMath
E = EllipticCurve("ep1")
E.isogeny_class()
Elliptic curves in class 91728ep
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 91728.v2 | 91728ep1 | \([0, 0, 0, -51744, -4538576]\) | \(-43614208/91\) | \(-31968120877056\) | \([]\) | \(331776\) | \(1.4769\) | \(\Gamma_0(N)\)-optimal |
| 91728.v3 | 91728ep2 | \([0, 0, 0, 89376, -22517264]\) | \(224755712/753571\) | \(-264728008982900736\) | \([]\) | \(995328\) | \(2.0262\) | |
| 91728.v1 | 91728ep3 | \([0, 0, 0, -827904, 714608944]\) | \(-178643795968/524596891\) | \(-184289855200173305856\) | \([]\) | \(2985984\) | \(2.5755\) |
Rank
sage: E.rank()
The elliptic curves in class 91728ep have rank \(0\).
Complex multiplication
The elliptic curves in class 91728ep do not have complex multiplication.Modular form 91728.2.a.ep
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
