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SageMath
E = EllipticCurve("bx1")
E.isogeny_class()
Elliptic curves in class 16170.bx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 16170.bx1 | 16170bw3 | \([1, 0, 0, -4371335571, -111242504032479]\) | \(78519570041710065450485106721/96428056919040\) | \(11344664468468136960\) | \([2]\) | \(8847360\) | \(3.8304\) | |
| 16170.bx2 | 16170bw5 | \([1, 0, 0, -1285691891, 16240774020225]\) | \(1997773216431678333214187041/187585177195046990066400\) | \(22069208511820083334321893600\) | \([2]\) | \(17694720\) | \(4.1770\) | |
| 16170.bx3 | 16170bw4 | \([1, 0, 0, -285503891, -1573174372575]\) | \(21876183941534093095979041/3572502915711058560000\) | \(420301395530490328525440000\) | \([2, 2]\) | \(8847360\) | \(3.8304\) | |
| 16170.bx4 | 16170bw2 | \([1, 0, 0, -273210771, -1738150501599]\) | \(19170300594578891358373921/671785075055001600\) | \(79034842295145883238400\) | \([2, 2]\) | \(4423680\) | \(3.4838\) | |
| 16170.bx5 | 16170bw1 | \([1, 0, 0, -16309651, -29706673375]\) | \(-4078208988807294650401/880065599546327040\) | \(-103538837721025829928960\) | \([2]\) | \(2211840\) | \(3.1372\) | \(\Gamma_0(N)\)-optimal |
| 16170.bx6 | 16170bw6 | \([1, 0, 0, 517994189, -8828601335359]\) | \(130650216943167617311657439/361816948816603087500000\) | \(-42567402211324536641287500000\) | \([2]\) | \(17694720\) | \(4.1770\) |
Rank
sage: E.rank()
The elliptic curves in class 16170.bx have rank \(0\).
Complex multiplication
The elliptic curves in class 16170.bx do not have complex multiplication.Modular form 16170.2.a.bx
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.
