Show commands:
SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 22050cn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22050.y4 | 22050cn1 | \([1, -1, 0, -1332, -25124]\) | \(-24389/12\) | \(-128649181500\) | \([2]\) | \(23040\) | \(0.83699\) | \(\Gamma_0(N)\)-optimal |
| 22050.y2 | 22050cn2 | \([1, -1, 0, -23382, -1370174]\) | \(131872229/18\) | \(192973772250\) | \([2]\) | \(46080\) | \(1.1836\) | |
| 22050.y3 | 22050cn3 | \([1, -1, 0, -12357, 2543701]\) | \(-19465109/248832\) | \(-2667669427584000\) | \([2]\) | \(115200\) | \(1.6417\) | |
| 22050.y1 | 22050cn4 | \([1, -1, 0, -365157, 84746101]\) | \(502270291349/1889568\) | \(20257614715716000\) | \([2]\) | \(230400\) | \(1.9883\) |
Rank
sage: E.rank()
The elliptic curves in class 22050cn have rank \(0\).
Complex multiplication
The elliptic curves in class 22050cn do not have complex multiplication.Modular form 22050.2.a.cn
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
