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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 142830bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
142830.cp2 | 142830bc1 | \([1, -1, 1, 3868, 1226759]\) | \(1601613/163840\) | \(-654863401451520\) | \([]\) | \(653400\) | \(1.5224\) | \(\Gamma_0(N)\)-optimal |
142830.cp1 | 142830bc2 | \([1, -1, 1, -757892, 254198791]\) | \(-16522921323/4000\) | \(-11655161612748000\) | \([]\) | \(1960200\) | \(2.0717\) |
Rank
sage: E.rank()
The elliptic curves in class 142830bc have rank \(0\).
Complex multiplication
The elliptic curves in class 142830bc do not have complex multiplication.Modular form 142830.2.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.