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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 2450c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2450.b1 | 2450c1 | \([1, -1, 0, -161317, 24978841]\) | \(-5154200289/20\) | \(-1801500312500\) | \([]\) | \(20160\) | \(1.5648\) | \(\Gamma_0(N)\)-optimal |
2450.b2 | 2450c2 | \([1, -1, 0, 1124933, -236901659]\) | \(1747829720511/1280000000\) | \(-115296020000000000000\) | \([]\) | \(141120\) | \(2.5378\) |
Rank
sage: E.rank()
The elliptic curves in class 2450c have rank \(1\).
Complex multiplication
The elliptic curves in class 2450c do not have complex multiplication.Modular form 2450.2.a.c
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.