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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 12870.bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12870.bp1 | 12870bw4 | \([1, -1, 1, -20444153, 35584770521]\) | \(1296294060988412126189641/647824320\) | \(472263929280\) | \([6]\) | \(331776\) | \(2.4762\) | |
| 12870.bp2 | 12870bw3 | \([1, -1, 1, -1277753, 556257881]\) | \(-316472948332146183241/7074906009600\) | \(-5157606480998400\) | \([6]\) | \(165888\) | \(2.1296\) | |
| 12870.bp3 | 12870bw2 | \([1, -1, 1, -252878, 48674081]\) | \(2453170411237305241/19353090685500\) | \(14108403109729500\) | \([2]\) | \(110592\) | \(1.9269\) | |
| 12870.bp4 | 12870bw1 | \([1, -1, 1, -5378, 1748081]\) | \(-23592983745241/1794399750000\) | \(-1308117417750000\) | \([2]\) | \(55296\) | \(1.5803\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12870.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 12870.bp do not have complex multiplication.Modular form 12870.2.a.bp
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
