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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 36225.bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 36225.bp1 | 36225bi4 | \([1, -1, 0, -966042, -365220509]\) | \(8753151307882969/65205\) | \(742725703125\) | \([2]\) | \(270336\) | \(1.8724\) | |
| 36225.bp2 | 36225bi2 | \([1, -1, 0, -60417, -5687384]\) | \(2141202151369/5832225\) | \(66432687890625\) | \([2, 2]\) | \(135168\) | \(1.5258\) | |
| 36225.bp3 | 36225bi3 | \([1, -1, 0, -36792, -10199759]\) | \(-483551781049/3672913125\) | \(-41836776064453125\) | \([2]\) | \(270336\) | \(1.8724\) | |
| 36225.bp4 | 36225bi1 | \([1, -1, 0, -5292, -9509]\) | \(1439069689/828345\) | \(9435367265625\) | \([2]\) | \(67584\) | \(1.1792\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 36225.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 36225.bp do not have complex multiplication.Modular form 36225.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
