Properties

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

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

Elliptic curves in class 126.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
126.a1 126b3 \([1, -1, 0, -12096, -509036]\) \(268498407453697/252\) \(183708\) \([2]\) \(128\) \(0.73841\)  
126.a2 126b5 \([1, -1, 0, -8226, 286474]\) \(84448510979617/933897762\) \(680811468498\) \([2]\) \(256\) \(1.0850\)  
126.a3 126b4 \([1, -1, 0, -936, -3668]\) \(124475734657/63011844\) \(45935634276\) \([2, 2]\) \(128\) \(0.73841\)  
126.a4 126b2 \([1, -1, 0, -756, -7808]\) \(65597103937/63504\) \(46294416\) \([2, 2]\) \(64\) \(0.39183\)  
126.a5 126b1 \([1, -1, 0, -36, -176]\) \(-7189057/16128\) \(-11757312\) \([2]\) \(32\) \(0.045260\) \(\Gamma_0(N)\)-optimal
126.a6 126b6 \([1, -1, 0, 3474, -31010]\) \(6359387729183/4218578658\) \(-3075343841682\) \([2]\) \(256\) \(1.0850\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 126.a do not have complex multiplication.

Modular form 126.2.a.a

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.