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SageMath
E = EllipticCurve("dk1")
E.isogeny_class()
Elliptic curves in class 129360.dk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129360.dk1 | 129360br4 | \([0, -1, 0, -798440120, -7690851577968]\) | \(233632133015204766393938/29145526885986328125\) | \(7022473405664062500000000000\) | \([2]\) | \(94371840\) | \(4.0729\) | |
| 129360.dk2 | 129360br2 | \([0, -1, 0, -199438640, 959688195600]\) | \(7282213870869695463556/912102595400390625\) | \(109883349244170810000000000\) | \([2, 2]\) | \(47185920\) | \(3.7263\) | |
| 129360.dk3 | 129360br1 | \([0, -1, 0, -193008860, 1032128668992]\) | \(26401417552259125806544/507547744790625\) | \(15286396064479293600000\) | \([2]\) | \(23592960\) | \(3.3797\) | \(\Gamma_0(N)\)-optimal |
| 129360.dk4 | 129360br3 | \([0, -1, 0, 296686360, 4973934795600]\) | \(11986661998777424518222/51295853620928503125\) | \(-12359487247664368566585600000\) | \([2]\) | \(94371840\) | \(4.0729\) |
Rank
sage: E.rank()
The elliptic curves in class 129360.dk have rank \(0\).
Complex multiplication
The elliptic curves in class 129360.dk do not have complex multiplication.Modular form 129360.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
