Properties

Label 25410v
Number of curves $8$
Conductor $25410$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("25410.w1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 25410v

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25410.w7 25410v1 [1, 0, 1, -4964, 46946] [2] 69120 \(\Gamma_0(N)\)-optimal
25410.w5 25410v2 [1, 0, 1, -43684, -3484318] [2, 2] 138240  
25410.w4 25410v3 [1, 0, 1, -324404, 71090402] [2] 207360  
25410.w6 25410v4 [1, 0, 1, -9804, -8742494] [2] 276480  
25410.w2 25410v5 [1, 0, 1, -697084, -224072158] [2] 276480  
25410.w3 25410v6 [1, 0, 1, -326824, 69975266] [2, 2] 414720  
25410.w8 25410v7 [1, 0, 1, 88206, 235655242] [2] 829440  
25410.w1 25410v8 [1, 0, 1, -780574, -167063734] [2] 829440  

Rank

sage: E.rank()
 

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

Modular form 25410.2.a.w

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

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 Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.