Properties

Label 32a
Number of curves 4
Conductor 32
CM -4
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("32.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 32a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
32.a4 32a1 [0, 0, 0, 4, 0] [4] 1 \(\Gamma_0(N)\)-optimal
32.a3 32a2 [0, 0, 0, -1, 0] [2, 2] 2  
32.a1 32a3 [0, 0, 0, -11, -14] [2] 4  
32.a2 32a4 [0, 0, 0, -11, 14] [4] 4  

Rank

sage: E.rank()
 

The elliptic curves in class 32a have rank \(0\).

Modular form 32.2.a.a

sage: E.q_eigenform(10)
 
\( q - 2q^{5} - 3q^{9} + 6q^{13} + 2q^{17} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.