Properties

Label 39a
Number of curves $4$
Conductor $39$
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 39a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
39.a3 39a1 \([1, 1, 0, -4, -5]\) \(10218313/1521\) \(1521\) \([2, 2]\) \(2\) \(-0.66494\) \(\Gamma_0(N)\)-optimal
39.a1 39a2 \([1, 1, 0, -69, -252]\) \(37159393753/1053\) \(1053\) \([2]\) \(4\) \(-0.31837\)  
39.a2 39a3 \([1, 1, 0, -19, 22]\) \(822656953/85683\) \(85683\) \([4]\) \(4\) \(-0.31837\)  
39.a4 39a4 \([1, 1, 0, 1, 0]\) \(12167/39\) \(-39\) \([2]\) \(4\) \(-1.0115\)  

Rank

sage: E.rank()
 

The elliptic curves in class 39a have rank \(0\).

Complex multiplication

The elliptic curves in class 39a do not have complex multiplication.

Modular form 39.2.a.a

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} - q^{4} + 2q^{5} - q^{6} - 4q^{7} - 3q^{8} + q^{9} + 2q^{10} + 4q^{11} + q^{12} + q^{13} - 4q^{14} - 2q^{15} - q^{16} + 2q^{17} + q^{18} + O(q^{20})\)  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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.