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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 75712.w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 75712.w1 | 75712f3 | \([0, -1, 0, -169411233, -848657634527]\) | \(-424962187484640625/182\) | \(-230288061366272\) | \([]\) | \(3483648\) | \(3.0027\) | |
| 75712.w2 | 75712f2 | \([0, -1, 0, -2087713, -1167942751]\) | \(-795309684625/6028568\) | \(-7628061744696393728\) | \([]\) | \(1161216\) | \(2.4533\) | |
| 75712.w3 | 75712f1 | \([0, -1, 0, 75487, -8554079]\) | \(37595375/46592\) | \(-58953743709765632\) | \([]\) | \(387072\) | \(1.9040\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 75712.w have rank \(1\).
Complex multiplication
The elliptic curves in class 75712.w do not have complex multiplication.Modular form 75712.2.a.w
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
