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SageMath
E = EllipticCurve("kp1")
E.isogeny_class()
Elliptic curves in class 458640kp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
458640.kp2 | 458640kp1 | \([0, 0, 0, -147, -445214]\) | \(-4/975\) | \(-85628895206400\) | \([2]\) | \(1105920\) | \(1.3523\) | \(\Gamma_0(N)\)-optimal* |
458640.kp1 | 458640kp2 | \([0, 0, 0, -88347, -9953174]\) | \(434163602/7605\) | \(1335810765219840\) | \([2]\) | \(2211840\) | \(1.6989\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 458640kp have rank \(0\).
Complex multiplication
The elliptic curves in class 458640kp do not have complex multiplication.Modular form 458640.2.a.kp
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.