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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 3330.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3330.i1 | 3330c2 | \([1, -1, 0, -5739, -165907]\) | \(1062144635427/54760\) | \(1077841080\) | \([2]\) | \(3456\) | \(0.80252\) | |
3330.i2 | 3330c1 | \([1, -1, 0, -339, -2827]\) | \(-219256227/59200\) | \(-1165233600\) | \([2]\) | \(1728\) | \(0.45594\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3330.i have rank \(1\).
Complex multiplication
The elliptic curves in class 3330.i do not have complex multiplication.Modular form 3330.2.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 LMFDB 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 LMFDB labels.