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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 75712.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 75712.f1 | 75712db3 | \([0, 1, 0, -79317, 21164521]\) | \(-178643795968/524596891\) | \(-162056255670452416\) | \([]\) | \(870912\) | \(1.9892\) | |
| 75712.f2 | 75712db1 | \([0, 1, 0, -4957, -136239]\) | \(-43614208/91\) | \(-28111335616\) | \([]\) | \(96768\) | \(0.89054\) | \(\Gamma_0(N)\)-optimal |
| 75712.f3 | 75712db2 | \([0, 1, 0, 8563, -664871]\) | \(224755712/753571\) | \(-232789970236096\) | \([]\) | \(290304\) | \(1.4398\) |
Rank
sage: E.rank()
The elliptic curves in class 75712.f have rank \(1\).
Complex multiplication
The elliptic curves in class 75712.f do not have complex multiplication.Modular form 75712.2.a.f
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.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
