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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 7920c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7920.f6 | 7920c1 | \([0, 0, 0, 582, -4417]\) | \(1869154304/1804275\) | \(-21045063600\) | \([2]\) | \(4096\) | \(0.66790\) | \(\Gamma_0(N)\)-optimal |
| 7920.f5 | 7920c2 | \([0, 0, 0, -3063, -40138]\) | \(17029316176/6125625\) | \(1143188640000\) | \([2, 2]\) | \(8192\) | \(1.0145\) | |
| 7920.f2 | 7920c3 | \([0, 0, 0, -43563, -3498838]\) | \(12247559771044/3294225\) | \(2459125785600\) | \([2, 2]\) | \(16384\) | \(1.3610\) | |
| 7920.f4 | 7920c4 | \([0, 0, 0, -20883, 1132418]\) | \(1349195526724/38671875\) | \(28868400000000\) | \([2]\) | \(16384\) | \(1.3610\) | |
| 7920.f1 | 7920c5 | \([0, 0, 0, -696963, -223955998]\) | \(25078144523224322/1815\) | \(2709780480\) | \([2]\) | \(32768\) | \(1.7076\) | |
| 7920.f3 | 7920c6 | \([0, 0, 0, -38163, -4398478]\) | \(-4117122162722/3215383215\) | \(-4800541416929280\) | \([2]\) | \(32768\) | \(1.7076\) |
Rank
sage: E.rank()
The elliptic curves in class 7920c have rank \(0\).
Complex multiplication
The elliptic curves in class 7920c do not have complex multiplication.Modular form 7920.2.a.c
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
