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Properties

Label 25886a
Number of curves 6
Conductor 25886
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("25886.d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 25886a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25886.d5 25886a1 [1, 1, 1, -963, -23683] [2] 26208 \(\Gamma_0(N)\)-optimal
25886.d4 25886a2 [1, 1, 1, -19453, -1051727] [2] 52416  
25886.d6 25886a3 [1, 1, 1, 8282, 490339] [2] 78624  
25886.d3 25886a4 [1, 1, 1, -65678, 5282947] [2] 157248  
25886.d2 25886a5 [1, 1, 1, -315293, 68208115] [2] 235872  
25886.d1 25886a6 [1, 1, 1, -5048733, 4364278259] [2] 471744  

Rank

sage: E.rank()
 

The elliptic curves in class 25886a have rank \(1\).

Modular form 25886.2.a.d

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.