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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 122010s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
122010.q4 | 122010s1 | \([1, 0, 1, -24134, 866072]\) | \(13212881163721/4879284480\) | \(574042939787520\) | \([2]\) | \(835584\) | \(1.5321\) | \(\Gamma_0(N)\)-optimal |
122010.q2 | 122010s2 | \([1, 0, 1, -341654, 76816856]\) | \(37488077912343241/10936976400\) | \(1286724336483600\) | \([2, 2]\) | \(1671168\) | \(1.8786\) | |
122010.q3 | 122010s3 | \([1, 0, 1, -297554, 97385096]\) | \(-24764567772437641/20510537169780\) | \(-2413044187487447220\) | \([2]\) | \(3342336\) | \(2.2252\) | |
122010.q1 | 122010s4 | \([1, 0, 1, -5466074, 4918368872]\) | \(153518910112934762761/13072500\) | \(1537966552500\) | \([2]\) | \(3342336\) | \(2.2252\) |
Rank
sage: E.rank()
The elliptic curves in class 122010s have rank \(1\).
Complex multiplication
The elliptic curves in class 122010s do not have complex multiplication.Modular form 122010.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.