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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 1290.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1290.k1 | 1290j2 | \([1, 1, 1, -1336, 18239]\) | \(263732349218689/4160250\) | \(4160250\) | \([2]\) | \(576\) | \(0.40401\) | |
1290.k2 | 1290j1 | \([1, 1, 1, -86, 239]\) | \(70393838689/8062500\) | \(8062500\) | \([2]\) | \(288\) | \(0.057441\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1290.k have rank \(0\).
Complex multiplication
The elliptic curves in class 1290.k do not have complex multiplication.Modular form 1290.2.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 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.