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SageMath
E = EllipticCurve("gs1")
E.isogeny_class()
Elliptic curves in class 100800.gs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 100800.gs1 | 100800dy6 | \([0, 0, 0, -241920300, 1448293502000]\) | \(524388516989299201/3150\) | \(9405849600000000\) | \([2]\) | \(9437184\) | \(3.1305\) | |
| 100800.gs2 | 100800dy4 | \([0, 0, 0, -15120300, 22628702000]\) | \(128031684631201/9922500\) | \(29628426240000000000\) | \([2, 2]\) | \(4718592\) | \(2.7839\) | |
| 100800.gs3 | 100800dy5 | \([0, 0, 0, -14112300, 25775678000]\) | \(-104094944089921/35880468750\) | \(-107138505600000000000000\) | \([2]\) | \(9437184\) | \(3.1305\) | |
| 100800.gs4 | 100800dy3 | \([0, 0, 0, -5328300, -4474402000]\) | \(5602762882081/345888060\) | \(1032816212951040000000\) | \([2]\) | \(4718592\) | \(2.7839\) | |
| 100800.gs5 | 100800dy2 | \([0, 0, 0, -1008300, 303518000]\) | \(37966934881/8643600\) | \(25809651302400000000\) | \([2, 2]\) | \(2359296\) | \(2.4374\) | |
| 100800.gs6 | 100800dy1 | \([0, 0, 0, 143700, 29342000]\) | \(109902239/188160\) | \(-561842749440000000\) | \([2]\) | \(1179648\) | \(2.0908\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 100800.gs have rank \(0\).
Complex multiplication
The elliptic curves in class 100800.gs do not have complex multiplication.Modular form 100800.2.a.gs
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.
