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SageMath
E = EllipticCurve("fg1")
E.isogeny_class()
Elliptic curves in class 20160.fg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20160.fg1 | 20160fg5 | \([0, 0, 0, -377292, -89197936]\) | \(62161150998242/1607445\) | \(153593761628160\) | \([2]\) | \(131072\) | \(1.8288\) | |
| 20160.fg2 | 20160fg3 | \([0, 0, 0, -24492, -1280176]\) | \(34008619684/4862025\) | \(232286861721600\) | \([2, 2]\) | \(65536\) | \(1.4822\) | |
| 20160.fg3 | 20160fg2 | \([0, 0, 0, -6492, 181424]\) | \(2533446736/275625\) | \(3292047360000\) | \([2, 2]\) | \(32768\) | \(1.1356\) | |
| 20160.fg4 | 20160fg1 | \([0, 0, 0, -6312, 193016]\) | \(37256083456/525\) | \(391910400\) | \([2]\) | \(16384\) | \(0.78907\) | \(\Gamma_0(N)\)-optimal |
| 20160.fg5 | 20160fg4 | \([0, 0, 0, 8628, 901136]\) | \(1486779836/8203125\) | \(-391910400000000\) | \([2]\) | \(65536\) | \(1.4822\) | |
| 20160.fg6 | 20160fg6 | \([0, 0, 0, 40308, -6904816]\) | \(75798394558/259416045\) | \(-24787589110824960\) | \([2]\) | \(131072\) | \(1.8288\) |
Rank
sage: E.rank()
The elliptic curves in class 20160.fg have rank \(1\).
Complex multiplication
The elliptic curves in class 20160.fg do not have complex multiplication.Modular form 20160.2.a.fg
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
