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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 25200.bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25200.bp1 | 25200u4 | \([0, 0, 0, -34275, 2439250]\) | \(381775972/567\) | \(6613488000000\) | \([2]\) | \(65536\) | \(1.3610\) | |
| 25200.bp2 | 25200u2 | \([0, 0, 0, -2775, 13750]\) | \(810448/441\) | \(1285956000000\) | \([2, 2]\) | \(32768\) | \(1.0144\) | |
| 25200.bp3 | 25200u1 | \([0, 0, 0, -1650, -25625]\) | \(2725888/21\) | \(3827250000\) | \([2]\) | \(16384\) | \(0.66784\) | \(\Gamma_0(N)\)-optimal |
| 25200.bp4 | 25200u3 | \([0, 0, 0, 10725, 108250]\) | \(11696828/7203\) | \(-84015792000000\) | \([2]\) | \(65536\) | \(1.3610\) |
Rank
sage: E.rank()
The elliptic curves in class 25200.bp have rank \(0\).
Complex multiplication
The elliptic curves in class 25200.bp do not have complex multiplication.Modular form 25200.2.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.
