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SageMath
sage: E = EllipticCurve("k1")
sage: E.isogeny_class()
Elliptic curves in class 4719.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 4719.k1 | 4719j3 | [1, 0, 1, -830547, -291405149] | [2] | 30720 | |
| 4719.k2 | 4719j5 | [1, 0, 1, -364092, 81851779] | [2] | 61440 | |
| 4719.k3 | 4719j4 | [1, 0, 1, -57357, -3543245] | [2, 2] | 30720 | |
| 4719.k4 | 4719j2 | [1, 0, 1, -51912, -4556015] | [2, 2] | 15360 | |
| 4719.k5 | 4719j1 | [1, 0, 1, -2907, -86759] | [2] | 7680 | \(\Gamma_0(N)\)-optimal |
| 4719.k6 | 4719j6 | [1, 0, 1, 162258, -24099209] | [2] | 61440 |
Rank
sage: E.rank()
The elliptic curves in class 4719.k have rank \(1\).
Complex multiplication
The elliptic curves in class 4719.k do not have complex multiplication.Modular form 4719.2.a.k
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
