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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 121680ck
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121680.cf4 | 121680ck1 | \([0, 0, 0, 15717, -206518]\) | \(804357/500\) | \(-266903230464000\) | \([2]\) | \(414720\) | \(1.4573\) | \(\Gamma_0(N)\)-optimal |
| 121680.cf3 | 121680ck2 | \([0, 0, 0, -65403, -1682902]\) | \(57960603/31250\) | \(16681451904000000\) | \([2]\) | \(829440\) | \(1.8038\) | |
| 121680.cf2 | 121680ck3 | \([0, 0, 0, -187083, 35472762]\) | \(-1860867/320\) | \(-124526371205283840\) | \([2]\) | \(1244160\) | \(2.0066\) | |
| 121680.cf1 | 121680ck4 | \([0, 0, 0, -3107403, 2108315898]\) | \(8527173507/200\) | \(77828982003302400\) | \([2]\) | \(2488320\) | \(2.3531\) |
Rank
sage: E.rank()
The elliptic curves in class 121680ck have rank \(0\).
Complex multiplication
The elliptic curves in class 121680ck do not have complex multiplication.Modular form 121680.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
