Properties

Label 25410.bl
Number of curves 8
Conductor 25410
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("25410.bl1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 25410.bl

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25410.bl1 25410bj8 [1, 0, 1, -232416803, 1363776582848] [2] 2621440  
25410.bl2 25410bj6 [1, 0, 1, -14526053, 21308093948] [2, 2] 1310720  
25410.bl3 25410bj7 [1, 0, 1, -14435303, 21587495048] [2] 2621440  
25410.bl4 25410bj4 [1, 0, 1, -1823473, -947648044] [2] 655360  
25410.bl5 25410bj3 [1, 0, 1, -913553, 328508948] [2, 2] 655360  
25410.bl6 25410bj2 [1, 0, 1, -129473, -10527244] [2, 2] 327680  
25410.bl7 25410bj1 [1, 0, 1, 25407, -1172492] [2] 163840 \(\Gamma_0(N)\)-optimal
25410.bl8 25410bj5 [1, 0, 1, 153667, 1050376556] [2] 1310720  

Rank

sage: E.rank()
 

The elliptic curves in class 25410.bl have rank \(1\).

Modular form 25410.2.a.bl

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + q^{7} - q^{8} + q^{9} - q^{10} + q^{12} + 2q^{13} - q^{14} + q^{15} + q^{16} - 2q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.