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SageMath
E = EllipticCurve("dk1")
E.isogeny_class()
Elliptic curves in class 76230dk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 76230.de6 | 76230dk1 | \([1, -1, 1, 10867, -612939]\) | \(109902239/188160\) | \(-243002613047040\) | \([2]\) | \(327680\) | \(1.4453\) | \(\Gamma_0(N)\)-optimal |
| 76230.de5 | 76230dk2 | \([1, -1, 1, -76253, -6293163]\) | \(37966934881/8643600\) | \(11162932536848400\) | \([2, 2]\) | \(655360\) | \(1.7919\) | |
| 76230.de4 | 76230dk3 | \([1, -1, 1, -402953, 93154317]\) | \(5602762882081/345888060\) | \(446703350349550140\) | \([2]\) | \(1310720\) | \(2.1384\) | |
| 76230.de2 | 76230dk4 | \([1, -1, 1, -1143473, -470320419]\) | \(128031684631201/9922500\) | \(12814590922402500\) | \([2, 2]\) | \(1310720\) | \(2.1384\) | |
| 76230.de3 | 76230dk5 | \([1, -1, 1, -1067243, -535786743]\) | \(-104094944089921/35880468750\) | \(-46338476103330468750\) | \([2]\) | \(2621440\) | \(2.4850\) | |
| 76230.de1 | 76230dk6 | \([1, -1, 1, -18295223, -30115405119]\) | \(524388516989299201/3150\) | \(4068124102350\) | \([2]\) | \(2621440\) | \(2.4850\) |
Rank
sage: E.rank()
The elliptic curves in class 76230dk have rank \(0\).
Complex multiplication
The elliptic curves in class 76230dk do not have complex multiplication.Modular form 76230.2.a.dk
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
