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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 726.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 726.d1 | 726e3 | \([1, 0, 1, -1217868, 517205302]\) | \(112763292123580561/1932612\) | \(3423740047332\) | \([2]\) | \(12000\) | \(1.9468\) | |
| 726.d2 | 726e4 | \([1, 0, 1, -1216658, 518284622]\) | \(-112427521449300721/466873642818\) | \(-827095137544298898\) | \([2]\) | \(24000\) | \(2.2934\) | |
| 726.d3 | 726e1 | \([1, 0, 1, -5448, -113258]\) | \(10091699281/2737152\) | \(4849031734272\) | \([2]\) | \(2400\) | \(1.1421\) | \(\Gamma_0(N)\)-optimal |
| 726.d4 | 726e2 | \([1, 0, 1, 13912, -732778]\) | \(168105213359/228637728\) | \(-405045682053408\) | \([2]\) | \(4800\) | \(1.4887\) |
Rank
sage: E.rank()
The elliptic curves in class 726.d have rank \(1\).
Complex multiplication
The elliptic curves in class 726.d do not have complex multiplication.Modular form 726.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
