Show commands:
SageMath
E = EllipticCurve("gf1")
E.isogeny_class()
Elliptic curves in class 283920.gf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 283920.gf1 | 283920gf5 | \([0, 1, 0, -45427256, 117833036244]\) | \(524388516989299201/3150\) | \(62277420441600\) | \([2]\) | \(14155776\) | \(2.7124\) | |
| 283920.gf2 | 283920gf3 | \([0, 1, 0, -2839256, 1840359444]\) | \(128031684631201/9922500\) | \(196173874391040000\) | \([2, 2]\) | \(7077888\) | \(2.3658\) | |
| 283920.gf3 | 283920gf6 | \([0, 1, 0, -2649976, 2096493140]\) | \(-104094944089921/35880468750\) | \(-709378742217600000000\) | \([2]\) | \(14155776\) | \(2.7124\) | |
| 283920.gf4 | 283920gf4 | \([0, 1, 0, -1000536, -364417260]\) | \(5602762882081/345888060\) | \(6838417821698211840\) | \([2]\) | \(7077888\) | \(2.3658\) | |
| 283920.gf5 | 283920gf2 | \([0, 1, 0, -189336, 24634260]\) | \(37966934881/8643600\) | \(170889241691750400\) | \([2, 2]\) | \(3538944\) | \(2.0192\) | |
| 283920.gf6 | 283920gf1 | \([0, 1, 0, 26984, 2396564]\) | \(109902239/188160\) | \(-3720037914378240\) | \([2]\) | \(1769472\) | \(1.6727\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 283920.gf have rank \(0\).
Complex multiplication
The elliptic curves in class 283920.gf do not have complex multiplication.Modular form 283920.2.a.gf
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
