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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 80688.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
80688.bc1 | 80688y1 | \([0, 1, 0, -2856, -64332]\) | \(-9129329/864\) | \(-243907559424\) | \([]\) | \(57600\) | \(0.92683\) | \(\Gamma_0(N)\)-optimal |
80688.bc2 | 80688y2 | \([0, 1, 0, 3704, 4330868]\) | \(19902511/28697814\) | \(-8101404830490624\) | \([]\) | \(288000\) | \(1.7315\) |
Rank
sage: E.rank()
The elliptic curves in class 80688.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 80688.bc do not have complex multiplication.Modular form 80688.2.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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.