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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1610d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1610.c3 | 1610d1 | \([1, 0, 0, -151, 681]\) | \(380920459249/12622400\) | \(12622400\) | \([6]\) | \(576\) | \(0.13561\) | \(\Gamma_0(N)\)-optimal |
| 1610.c4 | 1610d2 | \([1, 0, 0, 49, 2401]\) | \(12994449551/2489452840\) | \(-2489452840\) | \([6]\) | \(1152\) | \(0.48219\) | |
| 1610.c1 | 1610d3 | \([1, 0, 0, -1691, -26675]\) | \(534774372149809/5323062500\) | \(5323062500\) | \([2]\) | \(1728\) | \(0.68492\) | |
| 1610.c2 | 1610d4 | \([1, 0, 0, -441, -64925]\) | \(-9486391169809/1813439640250\) | \(-1813439640250\) | \([2]\) | \(3456\) | \(1.0315\) |
Rank
sage: E.rank()
The elliptic curves in class 1610d have rank \(1\).
Complex multiplication
The elliptic curves in class 1610d do not have complex multiplication.Modular form 1610.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
