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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 3757b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3757.b2 | 3757b1 | \([1, -1, 1, -211747, 37495370]\) | \(43499078731809/82055753\) | \(1980626399884457\) | \([2]\) | \(34560\) | \(1.8252\) | \(\Gamma_0(N)\)-optimal |
3757.b1 | 3757b2 | \([1, -1, 1, -3386412, 2399446130]\) | \(177930109857804849/634933\) | \(15325739097877\) | \([2]\) | \(69120\) | \(2.1718\) |
Rank
sage: E.rank()
The elliptic curves in class 3757b have rank \(1\).
Complex multiplication
The elliptic curves in class 3757b do not have complex multiplication.Modular form 3757.2.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.