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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 31410b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31410.b1 | 31410b1 | \([1, -1, 0, -408915, -100608075]\) | \(-10372797669976737841/7632630000000\) | \(-5564187270000000\) | \([]\) | \(310464\) | \(1.9553\) | \(\Gamma_0(N)\)-optimal |
31410.b2 | 31410b2 | \([1, -1, 0, 1641735, 5591578635]\) | \(671282315177095816559/18919046447754148470\) | \(-13791984860412774234630\) | \([]\) | \(2173248\) | \(2.9283\) |
Rank
sage: E.rank()
The elliptic curves in class 31410b have rank \(0\).
Complex multiplication
The elliptic curves in class 31410b do not have complex multiplication.Modular form 31410.2.a.b
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.