Properties

Label 231a
Number of curves 6
Conductor 231
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("231.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 231a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
231.a4 231a1 [1, 1, 1, -34, 62] [4] 20 \(\Gamma_0(N)\)-optimal
231.a3 231a2 [1, 1, 1, -39, 36] [2, 4] 40  
231.a2 231a3 [1, 1, 1, -284, -1924] [2, 2] 80  
231.a6 231a4 [1, 1, 1, 126, 432] [4] 80  
231.a1 231a5 [1, 1, 1, -4519, -118810] [2] 160  
231.a5 231a6 [1, 1, 1, 31, -5578] [2] 160  

Rank

sage: E.rank()
 

The elliptic curves in class 231a have rank \(0\).

Modular form 231.2.a.a

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} + q^{7} + 3q^{8} + q^{9} + 2q^{10} - q^{11} + q^{12} + 6q^{13} - q^{14} + 2q^{15} - q^{16} + 2q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.