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SageMath
E = EllipticCurve("nl1")
E.isogeny_class()
Elliptic curves in class 458640.nl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
458640.nl1 | 458640nl4 | \([0, 0, 0, -3669267, -2705310286]\) | \(31103978031362/195\) | \(34251558082560\) | \([2]\) | \(7077888\) | \(2.2026\) | |
458640.nl2 | 458640nl3 | \([0, 0, 0, -317667, -6778366]\) | \(20183398562/11567205\) | \(2031768173899376640\) | \([2]\) | \(7077888\) | \(2.2026\) | \(\Gamma_0(N)\)-optimal* |
458640.nl3 | 458640nl2 | \([0, 0, 0, -229467, -42217126]\) | \(15214885924/38025\) | \(3339526913049600\) | \([2, 2]\) | \(3538944\) | \(1.8561\) | \(\Gamma_0(N)\)-optimal* |
458640.nl4 | 458640nl1 | \([0, 0, 0, -8967, -1160026]\) | \(-3631696/24375\) | \(-535180595040000\) | \([2]\) | \(1769472\) | \(1.5095\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 458640.nl have rank \(0\).
Complex multiplication
The elliptic curves in class 458640.nl do not have complex multiplication.Modular form 458640.2.a.nl
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.