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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 331200s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
331200.s2 | 331200s1 | \([0, 0, 0, -41700, 5200000]\) | \(-171879616/150903\) | \(-7040530368000000\) | \([2]\) | \(2621440\) | \(1.7381\) | \(\Gamma_0(N)\)-optimal |
331200.s1 | 331200s2 | \([0, 0, 0, -770700, 260350000]\) | \(135638288072/42849\) | \(15993303552000000\) | \([2]\) | \(5242880\) | \(2.0847\) |
Rank
sage: E.rank()
The elliptic curves in class 331200s have rank \(1\).
Complex multiplication
The elliptic curves in class 331200s do not have complex multiplication.Modular form 331200.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.