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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 10647f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10647.d6 | 10647f1 | \([1, -1, 1, 1489, 4502]\) | \(103823/63\) | \(-221680856943\) | \([2]\) | \(9216\) | \(0.86627\) | \(\Gamma_0(N)\)-optimal |
| 10647.d5 | 10647f2 | \([1, -1, 1, -6116, 41006]\) | \(7189057/3969\) | \(13965893987409\) | \([2, 2]\) | \(18432\) | \(1.2128\) | |
| 10647.d3 | 10647f3 | \([1, -1, 1, -59351, -5516728]\) | \(6570725617/45927\) | \(161605344711447\) | \([2]\) | \(36864\) | \(1.5594\) | |
| 10647.d2 | 10647f4 | \([1, -1, 1, -74561, 7843736]\) | \(13027640977/21609\) | \(76036533931449\) | \([2, 2]\) | \(36864\) | \(1.5594\) | |
| 10647.d1 | 10647f5 | \([1, -1, 1, -1192496, 501523832]\) | \(53297461115137/147\) | \(517255332867\) | \([2]\) | \(73728\) | \(1.9060\) | |
| 10647.d4 | 10647f6 | \([1, -1, 1, -51746, 12717020]\) | \(-4354703137/17294403\) | \(-60854572656469683\) | \([2]\) | \(73728\) | \(1.9060\) |
Rank
sage: E.rank()
The elliptic curves in class 10647f have rank \(1\).
Complex multiplication
The elliptic curves in class 10647f do not have complex multiplication.Modular form 10647.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
