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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1521d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1521.e4 | 1521d1 | \([1, -1, 0, 729, 15952]\) | \(12167/39\) | \(-137231006679\) | \([2]\) | \(1344\) | \(0.82027\) | \(\Gamma_0(N)\)-optimal |
1521.e3 | 1521d2 | \([1, -1, 0, -6876, 190867]\) | \(10218313/1521\) | \(5352009260481\) | \([2, 2]\) | \(2688\) | \(1.1668\) | |
1521.e2 | 1521d3 | \([1, -1, 0, -29691, -1775786]\) | \(822656953/85683\) | \(301496521673763\) | \([2]\) | \(5376\) | \(1.5134\) | |
1521.e1 | 1521d4 | \([1, -1, 0, -105741, 13260820]\) | \(37159393753/1053\) | \(3705237180333\) | \([2]\) | \(5376\) | \(1.5134\) |
Rank
sage: E.rank()
The elliptic curves in class 1521d have rank \(0\).
Complex multiplication
The elliptic curves in class 1521d do not have complex multiplication.Modular form 1521.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.