Show commands:
SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 21840y
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21840.e3 | 21840y1 | \([0, -1, 0, -9896, -374160]\) | \(26168974809769/117411840\) | \(480918896640\) | \([2]\) | \(36864\) | \(1.0933\) | \(\Gamma_0(N)\)-optimal |
| 21840.e2 | 21840y2 | \([0, -1, 0, -15016, 60016]\) | \(91422999252649/52587662400\) | \(215399065190400\) | \([2, 2]\) | \(73728\) | \(1.4399\) | |
| 21840.e4 | 21840y3 | \([0, -1, 0, 59864, 419440]\) | \(5792335463322071/3372408585000\) | \(-13813385564160000\) | \([2]\) | \(147456\) | \(1.7865\) | |
| 21840.e1 | 21840y4 | \([0, -1, 0, -171816, 27405936]\) | \(136948444639063849/367281893160\) | \(1504386634383360\) | \([2]\) | \(147456\) | \(1.7865\) |
Rank
sage: E.rank()
The elliptic curves in class 21840y have rank \(0\).
Complex multiplication
The elliptic curves in class 21840y do not have complex multiplication.Modular form 21840.2.a.y
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.
