Show commands:
SageMath
E = EllipticCurve("ec1")
E.isogeny_class()
Elliptic curves in class 34650.ec
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 34650.ec1 | 34650dp4 | \([1, -1, 1, -7422980, -7782276103]\) | \(3971101377248209009/56495958750\) | \(643524280136718750\) | \([2]\) | \(1179648\) | \(2.5568\) | |
| 34650.ec2 | 34650dp2 | \([1, -1, 1, -477230, -114168103]\) | \(1055257664218129/115307784900\) | \(1313427737376562500\) | \([2, 2]\) | \(589824\) | \(2.2102\) | |
| 34650.ec3 | 34650dp1 | \([1, -1, 1, -112730, 12677897]\) | \(13908844989649/1980372240\) | \(22557677546250000\) | \([2]\) | \(294912\) | \(1.8637\) | \(\Gamma_0(N)\)-optimal |
| 34650.ec4 | 34650dp3 | \([1, -1, 1, 636520, -568578103]\) | \(2503876820718671/13702874328990\) | \(-156084302903651718750\) | \([2]\) | \(1179648\) | \(2.5568\) |
Rank
sage: E.rank()
The elliptic curves in class 34650.ec have rank \(1\).
Complex multiplication
The elliptic curves in class 34650.ec do not have complex multiplication.Modular form 34650.2.a.ec
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.
