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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 121242k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121242.r2 | 121242k1 | \([1, 0, 1, -10409, 963884]\) | \(-70393838689/186227712\) | \(-329913751698432\) | \([2]\) | \(1036800\) | \(1.4734\) | \(\Gamma_0(N)\)-optimal |
121242.r1 | 121242k2 | \([1, 0, 1, -223369, 40574444]\) | \(695718426450529/795171168\) | \(1408694229553248\) | \([2]\) | \(2073600\) | \(1.8199\) |
Rank
sage: E.rank()
The elliptic curves in class 121242k have rank \(1\).
Complex multiplication
The elliptic curves in class 121242k do not have complex multiplication.Modular form 121242.2.a.k
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.