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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 257600i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
257600.i1 | 257600i1 | \([0, 1, 0, -708, 6838]\) | \(39304000/1127\) | \(1127000000\) | \([2]\) | \(110592\) | \(0.51512\) | \(\Gamma_0(N)\)-optimal |
257600.i2 | 257600i2 | \([0, 1, 0, 167, 23463]\) | \(8000/3703\) | \(-236992000000\) | \([2]\) | \(221184\) | \(0.86170\) |
Rank
sage: E.rank()
The elliptic curves in class 257600i have rank \(1\).
Complex multiplication
The elliptic curves in class 257600i do not have complex multiplication.Modular form 257600.2.a.i
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.