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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 296450ek
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
296450.ek2 | 296450ek1 | \([1, 1, 0, -9825, -197875]\) | \(18865/8\) | \(44486028125000\) | \([]\) | \(1088640\) | \(1.3156\) | \(\Gamma_0(N)\)-optimal |
296450.ek1 | 296450ek2 | \([1, 1, 0, -683575, -217819125]\) | \(6352571665/2\) | \(11121507031250\) | \([]\) | \(3265920\) | \(1.8649\) |
Rank
sage: E.rank()
The elliptic curves in class 296450ek have rank \(0\).
Complex multiplication
The elliptic curves in class 296450ek do not have complex multiplication.Modular form 296450.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.