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SageMath
E = EllipticCurve("gm1")
E.isogeny_class()
Elliptic curves in class 462400gm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
462400.gm2 | 462400gm1 | \([0, 1, 0, 337167, -19903537]\) | \(27440/17\) | \(-2626167507200000000\) | \([]\) | \(3317760\) | \(2.2235\) | \(\Gamma_0(N)\)-optimal |
462400.gm1 | 462400gm2 | \([0, 1, 0, -5442833, -5065843537]\) | \(-115431760/4913\) | \(-758962409580800000000\) | \([]\) | \(9953280\) | \(2.7728\) |
Rank
sage: E.rank()
The elliptic curves in class 462400gm have rank \(0\).
Complex multiplication
The elliptic curves in class 462400gm do not have complex multiplication.Modular form 462400.2.a.gm
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.