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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 34914.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
34914.bc1 | 34914bh2 | \([1, 0, 0, -23494, -1387996]\) | \(117872434296791/2811072\) | \(34202313024\) | \([2]\) | \(82944\) | \(1.1324\) | |
34914.bc2 | 34914bh1 | \([1, 0, 0, -1414, -23452]\) | \(-25698491351/4460544\) | \(-54271438848\) | \([2]\) | \(41472\) | \(0.78584\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 34914.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 34914.bc do not have complex multiplication.Modular form 34914.2.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 LMFDB 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 LMFDB labels.