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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 465690.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
465690.d1 | 465690d3 | \([1, 1, 0, -21813793, 37069872697]\) | \(24400330024019218849/1495971679687500\) | \(70379305621948242187500\) | \([2]\) | \(58982400\) | \(3.1360\) | \(\Gamma_0(N)\)-optimal* |
465690.d2 | 465690d2 | \([1, 1, 0, -4117573, -2502414467]\) | \(164106655117491169/37546256250000\) | \(1766396703533006250000\) | \([2, 2]\) | \(29491200\) | \(2.7894\) | \(\Gamma_0(N)\)-optimal* |
465690.d3 | 465690d1 | \([1, 1, 0, -3857653, -2917714643]\) | \(134949649760741089/10588320000\) | \(498136842709920000\) | \([2]\) | \(14745600\) | \(2.4428\) | \(\Gamma_0(N)\)-optimal* |
465690.d4 | 465690d4 | \([1, 1, 0, 9419927, -15490291967]\) | \(1964918972535908831/3341561738407500\) | \(-157206715899272374507500\) | \([2]\) | \(58982400\) | \(3.1360\) |
Rank
sage: E.rank()
The elliptic curves in class 465690.d have rank \(0\).
Complex multiplication
The elliptic curves in class 465690.d do not have complex multiplication.Modular form 465690.2.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 LMFDB 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 LMFDB labels.