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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 25200.ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25200.ca1 | 25200bz1 | \([0, 0, 0, -255, 1150]\) | \(78608/21\) | \(489888000\) | \([2]\) | \(8192\) | \(0.37580\) | \(\Gamma_0(N)\)-optimal |
25200.ca2 | 25200bz2 | \([0, 0, 0, 645, 7450]\) | \(318028/441\) | \(-41150592000\) | \([2]\) | \(16384\) | \(0.72238\) |
Rank
sage: E.rank()
The elliptic curves in class 25200.ca have rank \(1\).
Complex multiplication
The elliptic curves in class 25200.ca do not have complex multiplication.Modular form 25200.2.a.ca
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.