Show commands:
SageMath
E = EllipticCurve("pw1")
E.isogeny_class()
Elliptic curves in class 286650pw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286650.pw3 | 286650pw1 | \([1, -1, 1, -77405, -7407403]\) | \(38272753/4368\) | \(5853537758250000\) | \([2]\) | \(2359296\) | \(1.7578\) | \(\Gamma_0(N)\)-optimal |
| 286650.pw2 | 286650pw2 | \([1, -1, 1, -297905, 54773597]\) | \(2181825073/298116\) | \(399503952000562500\) | \([2, 2]\) | \(4718592\) | \(2.1044\) | |
| 286650.pw1 | 286650pw3 | \([1, -1, 1, -4597655, 3795556097]\) | \(8020417344913/187278\) | \(250970431384968750\) | \([2]\) | \(9437184\) | \(2.4510\) | |
| 286650.pw4 | 286650pw4 | \([1, -1, 1, 473845, 290929097]\) | \(8780064047/32388174\) | \(-43403250785203968750\) | \([2]\) | \(9437184\) | \(2.4510\) |
Rank
sage: E.rank()
The elliptic curves in class 286650pw have rank \(0\).
Complex multiplication
The elliptic curves in class 286650pw do not have complex multiplication.Modular form 286650.2.a.pw
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.
