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SageMath
E = EllipticCurve("cy1")
E.isogeny_class()
Elliptic curves in class 20160.cy
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20160.cy1 | 20160ev5 | \([0, 0, 0, -9676812, -11586348016]\) | \(524388516989299201/3150\) | \(601974374400\) | \([2]\) | \(393216\) | \(2.3258\) | |
| 20160.cy2 | 20160ev3 | \([0, 0, 0, -604812, -181029616]\) | \(128031684631201/9922500\) | \(1896219279360000\) | \([2, 2]\) | \(196608\) | \(1.9792\) | |
| 20160.cy3 | 20160ev6 | \([0, 0, 0, -564492, -206205424]\) | \(-104094944089921/35880468750\) | \(-6856864358400000000\) | \([2]\) | \(393216\) | \(2.3258\) | |
| 20160.cy4 | 20160ev4 | \([0, 0, 0, -213132, 35795216]\) | \(5602762882081/345888060\) | \(66100237628866560\) | \([2]\) | \(196608\) | \(1.9792\) | |
| 20160.cy5 | 20160ev2 | \([0, 0, 0, -40332, -2428144]\) | \(37966934881/8643600\) | \(1651817683353600\) | \([2, 2]\) | \(98304\) | \(1.6326\) | |
| 20160.cy6 | 20160ev1 | \([0, 0, 0, 5748, -234736]\) | \(109902239/188160\) | \(-35957935964160\) | \([2]\) | \(49152\) | \(1.2861\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 20160.cy have rank \(0\).
Complex multiplication
The elliptic curves in class 20160.cy do not have complex multiplication.Modular form 20160.2.a.cy
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.
