Properties

Label 40931.a
Number of curves $3$
Conductor $40931$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 40931.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
40931.a1 40931a3 \([0, -1, 1, -29099460, -60409576035]\) \(-52893159101157376/11\) \(-566724117971\) \([]\) \(1110000\) \(2.5521\)  
40931.a2 40931a2 \([0, -1, 1, -38450, -5243475]\) \(-122023936/161051\) \(-8297407811213411\) \([]\) \(222000\) \(1.7474\)  
40931.a3 40931a1 \([0, -1, 1, -1240, 40345]\) \(-4096/11\) \(-566724117971\) \([]\) \(44400\) \(0.94271\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 40931.a have rank \(1\).

Complex multiplication

The elliptic curves in class 40931.a do not have complex multiplication.

Modular form 40931.2.a.a

sage: E.q_eigenform(10)
 
\(q + 2 q^{2} - q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{9} + 2 q^{10} - q^{11} - 2 q^{12} + 4 q^{13} + 4 q^{14} - q^{15} - 4 q^{16} + 2 q^{17} - 4 q^{18} + O(q^{20})\) Copy content Toggle raw display

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.

\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.