Properties

Label 40.a
Number of curves $4$
Conductor $40$
CM no
Rank $0$
Graph

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

Elliptic curves in class 40.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
40.a1 40a2 \([0, 0, 0, -107, -426]\) \(132304644/5\) \(5120\) \([2]\) \(4\) \(-0.20221\)  
40.a2 40a1 \([0, 0, 0, -7, -6]\) \(148176/25\) \(6400\) \([2, 2]\) \(2\) \(-0.54879\) \(\Gamma_0(N)\)-optimal
40.a3 40a3 \([0, 0, 0, -2, 1]\) \(55296/5\) \(80\) \([4]\) \(4\) \(-0.89536\)  
40.a4 40a4 \([0, 0, 0, 13, -34]\) \(237276/625\) \(-640000\) \([4]\) \(4\) \(-0.20221\)  

Rank

sage: E.rank()
 

The elliptic curves in class 40.a have rank \(0\).

Complex multiplication

The elliptic curves in class 40.a do not have complex multiplication.

Modular form 40.2.a.a

sage: E.q_eigenform(10)
 
\(q + q^{5} - 4q^{7} - 3q^{9} + 4q^{11} - 2q^{13} + 2q^{17} + 4q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.