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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 8400.bn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8400.bn1 | 8400cd5 | \([0, 1, 0, -313608, 67492788]\) | \(53297461115137/147\) | \(9408000000\) | \([2]\) | \(32768\) | \(1.5721\) | |
| 8400.bn2 | 8400cd4 | \([0, 1, 0, -19608, 1048788]\) | \(13027640977/21609\) | \(1382976000000\) | \([2, 2]\) | \(16384\) | \(1.2255\) | |
| 8400.bn3 | 8400cd3 | \([0, 1, 0, -15608, -751212]\) | \(6570725617/45927\) | \(2939328000000\) | \([2]\) | \(16384\) | \(1.2255\) | |
| 8400.bn4 | 8400cd6 | \([0, 1, 0, -13608, 1708788]\) | \(-4354703137/17294403\) | \(-1106841792000000\) | \([2]\) | \(32768\) | \(1.5721\) | |
| 8400.bn5 | 8400cd2 | \([0, 1, 0, -1608, 4788]\) | \(7189057/3969\) | \(254016000000\) | \([2, 2]\) | \(8192\) | \(0.87892\) | |
| 8400.bn6 | 8400cd1 | \([0, 1, 0, 392, 788]\) | \(103823/63\) | \(-4032000000\) | \([2]\) | \(4096\) | \(0.53235\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8400.bn have rank \(1\).
Complex multiplication
The elliptic curves in class 8400.bn do not have complex multiplication.Modular form 8400.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
