Properties

Label 25350cs
Number of curves $6$
Conductor $25350$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("25350.cz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 25350cs

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25350.cz6 25350cs1 [1, 0, 0, 63287, 2459417] [2] 258048 \(\Gamma_0(N)\)-optimal
25350.cz5 25350cs2 [1, 0, 0, -274713, 20373417] [2, 2] 516096  
25350.cz3 25350cs3 [1, 0, 0, -2387213, -1405564083] [2, 2] 1032192  
25350.cz2 25350cs4 [1, 0, 0, -3570213, 2594158917] [2] 1032192  
25350.cz4 25350cs5 [1, 0, 0, -485963, -3582495333] [2] 2064384  
25350.cz1 25350cs6 [1, 0, 0, -38088463, -90480182833] [2] 2064384  

Rank

sage: E.rank()
 

The elliptic curves in class 25350cs have rank \(1\).

Modular form 25350.2.a.cz

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} - 4q^{11} + q^{12} + q^{16} + 6q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.