Show commands:
SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 5850bn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5850.bk3 | 5850bn1 | \([1, -1, 1, -1355, -122853]\) | \(-24137569/561600\) | \(-6396975000000\) | \([2]\) | \(9216\) | \(1.1380\) | \(\Gamma_0(N)\)-optimal |
| 5850.bk2 | 5850bn2 | \([1, -1, 1, -46355, -3812853]\) | \(967068262369/4928040\) | \(56133455625000\) | \([2]\) | \(18432\) | \(1.4845\) | |
| 5850.bk4 | 5850bn3 | \([1, -1, 1, 12145, 3252147]\) | \(17394111071/411937500\) | \(-4692225585937500\) | \([2]\) | \(27648\) | \(1.6873\) | |
| 5850.bk1 | 5850bn4 | \([1, -1, 1, -269105, 51064647]\) | \(189208196468929/10860320250\) | \(123705835347656250\) | \([2]\) | \(55296\) | \(2.0338\) |
Rank
sage: E.rank()
The elliptic curves in class 5850bn have rank \(1\).
Complex multiplication
The elliptic curves in class 5850bn do not have complex multiplication.Modular form 5850.2.a.bn
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
