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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 80850ck
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 80850.cr4 | 80850ck1 | \([1, 0, 1, -118851, -57680402]\) | \(-100999381393/723148272\) | \(-1329338610195750000\) | \([2]\) | \(1179648\) | \(2.1608\) | \(\Gamma_0(N)\)-optimal |
| 80850.cr3 | 80850ck2 | \([1, 0, 1, -3083351, -2079469402]\) | \(1763535241378513/4612311396\) | \(8478653491062562500\) | \([2, 2]\) | \(2359296\) | \(2.5073\) | |
| 80850.cr2 | 80850ck3 | \([1, 0, 1, -4296101, -291875902]\) | \(4770223741048753/2740574865798\) | \(5037904568535451593750\) | \([2]\) | \(4718592\) | \(2.8539\) | |
| 80850.cr1 | 80850ck4 | \([1, 0, 1, -49302601, -133249700902]\) | \(7209828390823479793/49509306\) | \(91011255337406250\) | \([2]\) | \(4718592\) | \(2.8539\) |
Rank
sage: E.rank()
The elliptic curves in class 80850ck have rank \(0\).
Complex multiplication
The elliptic curves in class 80850ck do not have complex multiplication.Modular form 80850.2.a.ck
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
