Show commands:
SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 29232.bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 29232.bk1 | 29232ba4 | \([0, 0, 0, -29465859, 61563992002]\) | \(947531277805646290177/38367\) | \(114563248128\) | \([2]\) | \(786432\) | \(2.5331\) | |
| 29232.bk2 | 29232ba6 | \([0, 0, 0, -6115539, -4732031918]\) | \(8471112631466271697/1662662681263647\) | \(4964684163650349723648\) | \([2]\) | \(1572864\) | \(2.8797\) | |
| 29232.bk3 | 29232ba3 | \([0, 0, 0, -1876899, 923161570]\) | \(244883173420511137/18418027974129\) | \(54995936842301607936\) | \([2, 2]\) | \(786432\) | \(2.5331\) | |
| 29232.bk4 | 29232ba2 | \([0, 0, 0, -1841619, 961934290]\) | \(231331938231569617/1472026689\) | \(4395448140926976\) | \([2, 2]\) | \(393216\) | \(2.1865\) | |
| 29232.bk5 | 29232ba1 | \([0, 0, 0, -112899, 15632962]\) | \(-53297461115137/4513839183\) | \(-13478251579011072\) | \([2]\) | \(196608\) | \(1.8400\) | \(\Gamma_0(N)\)-optimal |
| 29232.bk6 | 29232ba5 | \([0, 0, 0, 1797261, 4096900978]\) | \(215015459663151503/2552757445339983\) | \(-7622492887666063798272\) | \([2]\) | \(1572864\) | \(2.8797\) |
Rank
sage: E.rank()
The elliptic curves in class 29232.bk have rank \(1\).
Complex multiplication
The elliptic curves in class 29232.bk do not have complex multiplication.Modular form 29232.2.a.bk
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
