Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 85782.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85782.a1 | 85782b6 | \([1, 1, 0, -23332721, -43390387461]\) | \(2361739090258884097/5202\) | \(3094270915842\) | \([2]\) | \(2867200\) | \(2.5307\) | |
| 85782.a2 | 85782b4 | \([1, 1, 0, -1458311, -678414495]\) | \(576615941610337/27060804\) | \(16096397304210084\) | \([2, 2]\) | \(1433600\) | \(2.1842\) | |
| 85782.a3 | 85782b5 | \([1, 1, 0, -1382621, -751879209]\) | \(-491411892194497/125563633938\) | \(-74688177735829468098\) | \([2]\) | \(2867200\) | \(2.5307\) | |
| 85782.a4 | 85782b2 | \([1, 1, 0, -95891, -9466275]\) | \(163936758817/30338064\) | \(18045787981190544\) | \([2, 2]\) | \(716800\) | \(1.8376\) | |
| 85782.a5 | 85782b1 | \([1, 1, 0, -28611, 1715661]\) | \(4354703137/352512\) | \(209682358532352\) | \([2]\) | \(358400\) | \(1.4910\) | \(\Gamma_0(N)\)-optimal |
| 85782.a6 | 85782b3 | \([1, 1, 0, 190049, -54816359]\) | \(1276229915423/2927177028\) | \(-1741153160949869988\) | \([2]\) | \(1433600\) | \(2.1842\) |
Rank
sage: E.rank()
The elliptic curves in class 85782.a have rank \(1\).
Complex multiplication
The elliptic curves in class 85782.a do not have complex multiplication.Modular form 85782.2.a.a
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
