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SageMath
E = EllipticCurve("kp1")
E.isogeny_class()
Elliptic curves in class 478800.kp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
478800.kp1 | 478800kp2 | \([0, 0, 0, -2805675, 1746668250]\) | \(1413487789441083/55278125000\) | \(95520600000000000000\) | \([2]\) | \(14155776\) | \(2.6010\) | \(\Gamma_0(N)\)-optimal* |
478800.kp2 | 478800kp1 | \([0, 0, 0, -453675, -80835750]\) | \(5976054062523/1824760000\) | \(3153185280000000000\) | \([2]\) | \(7077888\) | \(2.2544\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 478800.kp have rank \(0\).
Complex multiplication
The elliptic curves in class 478800.kp do not have complex multiplication.Modular form 478800.2.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 LMFDB 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 LMFDB labels.