Properties

Label 4998.bp
Number of curves $4$
Conductor $4998$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bp1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 4998.bp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4998.bp1 4998bl3 \([1, 0, 0, -248872, -47807992]\) \(14489843500598257/6246072\) \(734844124728\) \([2]\) \(36864\) \(1.6195\)  
4998.bp2 4998bl4 \([1, 0, 0, -33272, 1231992]\) \(34623662831857/14438442312\) \(1698668299564488\) \([2]\) \(36864\) \(1.6195\)  
4998.bp3 4998bl2 \([1, 0, 0, -15632, -740160]\) \(3590714269297/73410624\) \(8636686502976\) \([2, 2]\) \(18432\) \(1.2729\)  
4998.bp4 4998bl1 \([1, 0, 0, 48, -34560]\) \(103823/4386816\) \(-516104515584\) \([2]\) \(9216\) \(0.92636\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 4998.bp have rank \(0\).

Complex multiplication

The elliptic curves in class 4998.bp do not have complex multiplication.

Modular form 4998.2.a.bp

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} + q^{8} + q^{9} + 2q^{10} + q^{12} + 6q^{13} + 2q^{15} + q^{16} - q^{17} + q^{18} + O(q^{20})\)  Toggle raw display

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.