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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 348075.cp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
348075.cp1 | 348075cp2 | \([1, -1, 0, -13767, -234234]\) | \(684030715731/338005577\) | \(142596102796875\) | \([2]\) | \(983040\) | \(1.4090\) | |
348075.cp2 | 348075cp1 | \([1, -1, 0, -7392, 243891]\) | \(105890949891/1288651\) | \(543649640625\) | \([2]\) | \(491520\) | \(1.0624\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 348075.cp have rank \(1\).
Complex multiplication
The elliptic curves in class 348075.cp do not have complex multiplication.Modular form 348075.2.a.cp
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.