Show commands:
SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 9680g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 9680.s3 | 9680g1 | \([0, 0, 0, -610082, 183413131]\) | \(885956203616256/15125\) | \(428717762000\) | \([2]\) | \(69120\) | \(1.7739\) | \(\Gamma_0(N)\)-optimal |
| 9680.s2 | 9680g2 | \([0, 0, 0, -610687, 183031134]\) | \(55537159171536/228765625\) | \(103749698404000000\) | \([2, 2]\) | \(138240\) | \(2.1205\) | |
| 9680.s1 | 9680g3 | \([0, 0, 0, -913187, -17284366]\) | \(46424454082884/26794860125\) | \(48607978698654848000\) | \([2]\) | \(276480\) | \(2.4671\) | |
| 9680.s4 | 9680g4 | \([0, 0, 0, -317867, 358898826]\) | \(-1957960715364/29541015625\) | \(-53589720250000000000\) | \([2]\) | \(276480\) | \(2.4671\) |
Rank
sage: E.rank()
The elliptic curves in class 9680g have rank \(1\).
Complex multiplication
The elliptic curves in class 9680g do not have complex multiplication.Modular form 9680.2.a.g
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.
