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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 3870b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3870.b2 | 3870b1 | \([1, -1, 0, -75, 261]\) | \(1740992427/68800\) | \(1857600\) | \([2]\) | \(768\) | \(-0.030268\) | \(\Gamma_0(N)\)-optimal |
3870.b1 | 3870b2 | \([1, -1, 0, -195, -675]\) | \(30459021867/9245000\) | \(249615000\) | \([2]\) | \(1536\) | \(0.31631\) |
Rank
sage: E.rank()
The elliptic curves in class 3870b have rank \(1\).
Complex multiplication
The elliptic curves in class 3870b do not have complex multiplication.Modular form 3870.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.