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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 417450.t
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
417450.t1 | 417450t2 | \([1, 1, 0, -1021000, 6324424000]\) | \(-4252315368601/621860236800\) | \(-17213489733838200000000\) | \([]\) | \(29859840\) | \(2.9461\) | \(\Gamma_0(N)\)-optimal* |
417450.t2 | 417450t1 | \([1, 1, 0, 113375, -233397875]\) | \(5822285399/853875000\) | \(-23635807013671875000\) | \([]\) | \(9953280\) | \(2.3968\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 417450.t have rank \(1\).
Complex multiplication
The elliptic curves in class 417450.t do not have complex multiplication.Modular form 417450.2.a.t
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.