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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 5200.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5200.bc1 | 5200bj2 | \([0, -1, 0, -4532208, 3715262912]\) | \(-6434774386429585/140608\) | \(-224972800000000\) | \([]\) | \(129600\) | \(2.2803\) | |
5200.bc2 | 5200bj1 | \([0, -1, 0, -52208, 5822912]\) | \(-9836106385/3407872\) | \(-5452595200000000\) | \([]\) | \(43200\) | \(1.7309\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5200.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 5200.bc do not have complex multiplication.Modular form 5200.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.