Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("25410.ct1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 25410ct

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25410.ct7 25410ct1 [1, 0, 0, -60200, -5688000] [2] 138240 \(\Gamma_0(N)\)-optimal
25410.ct6 25410ct2 [1, 0, 0, -69880, -3738448] [2, 2] 276480  
25410.ct5 25410ct3 [1, 0, 0, -178175, 21985305] [2] 414720  
25410.ct8 25410ct4 [1, 0, 0, 232620, -27393948] [2] 552960  
25410.ct4 25410ct5 [1, 0, 0, -527260, 144727100] [2] 552960  
25410.ct2 25410ct6 [1, 0, 0, -2656255, 1665943577] [2, 2] 829440  
25410.ct3 25410ct7 [1, 0, 0, -2462655, 1919133657] [2] 1658880  
25410.ct1 25410ct8 [1, 0, 0, -42499135, 106635995225] [2] 1658880  

Rank

sage: E.rank()
 

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

Modular form 25410.2.a.ct

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

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.