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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 86190k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
86190.k1 | 86190k1 | \([1, 1, 0, -781628, 250778832]\) | \(4980061835533/313344000\) | \(3322856251533312000\) | \([2]\) | \(2515968\) | \(2.3050\) | \(\Gamma_0(N)\)-optimal |
86190.k2 | 86190k2 | \([1, 1, 0, 624452, 1055337808]\) | \(2539391358707/46818000000\) | \(-496481451645114000000\) | \([2]\) | \(5031936\) | \(2.6516\) |
Rank
sage: E.rank()
The elliptic curves in class 86190k have rank \(0\).
Complex multiplication
The elliptic curves in class 86190k do not have complex multiplication.Modular form 86190.2.a.k
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.