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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 85176.bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85176.bp1 | 85176o4 | \([0, 0, 0, -6133179, -5846238970]\) | \(7080974546692/189\) | \(681003592528896\) | \([2]\) | \(1474560\) | \(2.3600\) | |
| 85176.bp2 | 85176o3 | \([0, 0, 0, -596739, 21280142]\) | \(6522128932/3720087\) | \(13404193711746259968\) | \([2]\) | \(1474560\) | \(2.3600\) | |
| 85176.bp3 | 85176o2 | \([0, 0, 0, -383799, -91109590]\) | \(6940769488/35721\) | \(32177419746990336\) | \([2, 2]\) | \(737280\) | \(2.0134\) | |
| 85176.bp4 | 85176o1 | \([0, 0, 0, -11154, -2941783]\) | \(-2725888/64827\) | \(-3649753628709552\) | \([2]\) | \(368640\) | \(1.6668\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 85176.bp have rank \(0\).
Complex multiplication
The elliptic curves in class 85176.bp do not have complex multiplication.Modular form 85176.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
