Properties

Label 24640.bp
Number of curves $4$
Conductor $24640$
CM no
Rank $1$
Graph

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

Elliptic curves in class 24640.bp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
24640.bp1 24640o4 \([0, -1, 0, -225281, -40856575]\) \(4823468134087681/30382271150\) \(7964530088345600\) \([2]\) \(221184\) \(1.8885\)  
24640.bp2 24640o2 \([0, -1, 0, -17281, 842625]\) \(2177286259681/105875000\) \(27754496000000\) \([2]\) \(73728\) \(1.3392\)  
24640.bp3 24640o3 \([0, -1, 0, -5761, -1386879]\) \(-80677568161/3131816380\) \(-820986873118720\) \([2]\) \(110592\) \(1.5420\)  
24640.bp4 24640o1 \([0, -1, 0, 639, 50561]\) \(109902239/4312000\) \(-1130364928000\) \([2]\) \(36864\) \(0.99266\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 24640.bp have rank \(1\).

Complex multiplication

The elliptic curves in class 24640.bp do not have complex multiplication.

Modular form 24640.2.a.bp

sage: E.q_eigenform(10)
 
\(q + 2 q^{3} - q^{5} + q^{7} + q^{9} + q^{11} - 2 q^{13} - 2 q^{15} + 6 q^{17} - 2 q^{19} + O(q^{20})\) Copy content 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.

\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.