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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 32448.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32448.s1 | 32448cc2 | \([0, -1, 0, -4801, 713569]\) | \(-276301129/4782969\) | \(-211896699715584\) | \([]\) | \(86016\) | \(1.4301\) | |
32448.s2 | 32448cc1 | \([0, -1, 0, -641, -6111]\) | \(-658489/9\) | \(-398721024\) | \([]\) | \(12288\) | \(0.45713\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 32448.s have rank \(0\).
Complex multiplication
The elliptic curves in class 32448.s do not have complex multiplication.Modular form 32448.2.a.s
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.