Properties

Label 126.a
Number of curves $6$
Conductor $126$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("126.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 126.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
126.a1 126b3 [1, -1, 0, -12096, -509036] [2] 128  
126.a2 126b5 [1, -1, 0, -8226, 286474] [2] 256  
126.a3 126b4 [1, -1, 0, -936, -3668] [2, 2] 128  
126.a4 126b2 [1, -1, 0, -756, -7808] [2, 2] 64  
126.a5 126b1 [1, -1, 0, -36, -176] [2] 32 \(\Gamma_0(N)\)-optimal
126.a6 126b6 [1, -1, 0, 3474, -31010] [2] 256  

Rank

sage: E.rank()
 

The elliptic curves in class 126.a have rank \(0\).

Modular form 126.2.a.a

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.