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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 118580bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
118580.be1 | 118580bc1 | \([0, -1, 0, -3670, 87837]\) | \(-3937024/55\) | \(-76389710320\) | \([]\) | \(155520\) | \(0.89424\) | \(\Gamma_0(N)\)-optimal |
118580.be2 | 118580bc2 | \([0, -1, 0, 13270, 430025]\) | \(186050816/166375\) | \(-231078873718000\) | \([]\) | \(466560\) | \(1.4435\) |
Rank
sage: E.rank()
The elliptic curves in class 118580bc have rank \(1\).
Complex multiplication
The elliptic curves in class 118580bc do not have complex multiplication.Modular form 118580.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.