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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 67626bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 67626.bh4 | 67626bc1 | \([1, -1, 1, -50774, -91922587]\) | \(-822656953/207028224\) | \(-3642928212433895424\) | \([2]\) | \(1638400\) | \(2.2408\) | \(\Gamma_0(N)\)-optimal |
| 67626.bh3 | 67626bc2 | \([1, -1, 1, -3380054, -2369150107]\) | \(242702053576633/2554695936\) | \(44953164933901076736\) | \([2, 2]\) | \(3276800\) | \(2.5874\) | |
| 67626.bh2 | 67626bc3 | \([1, -1, 1, -6085094, 1961077925]\) | \(1416134368422073/725251155408\) | \(12761728058566945577808\) | \([2]\) | \(6553600\) | \(2.9340\) | |
| 67626.bh1 | 67626bc4 | \([1, -1, 1, -53943494, -152481890779]\) | \(986551739719628473/111045168\) | \(1953982735038395568\) | \([2]\) | \(6553600\) | \(2.9340\) |
Rank
sage: E.rank()
The elliptic curves in class 67626bc have rank \(0\).
Complex multiplication
The elliptic curves in class 67626bc do not have complex multiplication.Modular form 67626.2.a.bc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
