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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 17424.ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
17424.ca1 | 17424cc2 | \([0, 0, 0, -906411, -341164582]\) | \(-128667913/4096\) | \(-2621735685832310784\) | \([]\) | \(304128\) | \(2.3102\) | |
17424.ca2 | 17424cc1 | \([0, 0, 0, 51909, -1727638]\) | \(24167/16\) | \(-10241155022782464\) | \([]\) | \(101376\) | \(1.7609\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 17424.ca have rank \(0\).
Complex multiplication
The elliptic curves in class 17424.ca do not have complex multiplication.Modular form 17424.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.