Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 4719d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4719.c4 | 4719d1 | \([1, 1, 1, 58, 386]\) | \(12167/39\) | \(-69090879\) | \([2]\) | \(1280\) | \(0.18743\) | \(\Gamma_0(N)\)-optimal |
| 4719.c3 | 4719d2 | \([1, 1, 1, -547, 4016]\) | \(10218313/1521\) | \(2694544281\) | \([2, 2]\) | \(2560\) | \(0.53401\) | |
| 4719.c2 | 4719d3 | \([1, 1, 1, -2362, -40996]\) | \(822656953/85683\) | \(151792661163\) | \([2]\) | \(5120\) | \(0.88058\) | |
| 4719.c1 | 4719d4 | \([1, 1, 1, -8412, 293448]\) | \(37159393753/1053\) | \(1865453733\) | \([2]\) | \(5120\) | \(0.88058\) |
Rank
sage: E.rank()
The elliptic curves in class 4719d have rank \(0\).
Complex multiplication
The elliptic curves in class 4719d do not have complex multiplication.Modular form 4719.2.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.
