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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 52800ek
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
52800.u1 | 52800ek1 | \([0, -1, 0, -13133, -573363]\) | \(15657723904/49005\) | \(784080000000\) | \([2]\) | \(98304\) | \(1.1497\) | \(\Gamma_0(N)\)-optimal |
52800.u2 | 52800ek2 | \([0, -1, 0, -7633, -1062863]\) | \(-192143824/1804275\) | \(-461894400000000\) | \([2]\) | \(196608\) | \(1.4963\) |
Rank
sage: E.rank()
The elliptic curves in class 52800ek have rank \(2\).
Complex multiplication
The elliptic curves in class 52800ek do not have complex multiplication.Modular form 52800.2.a.ek
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.