Properties

Label 40425.cp
Number of curves $6$
Conductor $40425$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("cp1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 40425.cp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
40425.cp1 40425cg6 [1, 0, 1, -3735270026, 87867764877323] [2] 8847360  
40425.cp2 40425cg4 [1, 0, 1, -233454401, 1372918939823] [2, 2] 4423680  
40425.cp3 40425cg5 [1, 0, 1, -232296776, 1387208662823] [2] 8847360  
40425.cp4 40425cg3 [1, 0, 1, -31170151, -35325184177] [2] 4423680  
40425.cp5 40425cg2 [1, 0, 1, -14663276, 21227369573] [2, 2] 2211840  
40425.cp6 40425cg1 [1, 0, 1, 42849, 991741573] [2] 1105920 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 40425.cp have rank \(1\).

Complex multiplication

The elliptic curves in class 40425.cp do not have complex multiplication.

Modular form 40425.2.a.cp

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} - q^{4} + q^{6} - 3q^{8} + q^{9} - q^{11} - q^{12} - 2q^{13} - q^{16} + 2q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.