Show commands:
SageMath
E = EllipticCurve("cw1")
E.isogeny_class()
Elliptic curves in class 72450.cw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 72450.cw1 | 72450dg3 | \([1, -1, 1, -380480, 89647647]\) | \(534774372149809/5323062500\) | \(60633008789062500\) | \([2]\) | \(995328\) | \(2.0389\) | |
| 72450.cw2 | 72450dg4 | \([1, -1, 1, -99230, 219022647]\) | \(-9486391169809/1813439640250\) | \(-20656210902222656250\) | \([2]\) | \(1990656\) | \(2.3855\) | |
| 72450.cw3 | 72450dg1 | \([1, -1, 1, -33980, -2332353]\) | \(380920459249/12622400\) | \(143777025000000\) | \([2]\) | \(331776\) | \(1.4896\) | \(\Gamma_0(N)\)-optimal |
| 72450.cw4 | 72450dg2 | \([1, -1, 1, 11020, -8092353]\) | \(12994449551/2489452840\) | \(-28356423755625000\) | \([2]\) | \(663552\) | \(1.8362\) |
Rank
sage: E.rank()
The elliptic curves in class 72450.cw have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.cw do not have complex multiplication.Modular form 72450.2.a.cw
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
