Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 126.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
126.b1 126a6 \([1, -1, 1, -24575, 1488935]\) \(2251439055699625/25088\) \(18289152\) \([6]\) \(144\) \(0.96241\)  
126.b2 126a5 \([1, -1, 1, -1535, 23591]\) \(-548347731625/1835008\) \(-1337720832\) \([6]\) \(72\) \(0.61583\)  
126.b3 126a4 \([1, -1, 1, -320, 1883]\) \(4956477625/941192\) \(686128968\) \([6]\) \(48\) \(0.41310\)  
126.b4 126a2 \([1, -1, 1, -95, -331]\) \(128787625/98\) \(71442\) \([2]\) \(16\) \(-0.13621\)  
126.b5 126a1 \([1, -1, 1, -5, -7]\) \(-15625/28\) \(-20412\) \([2]\) \(8\) \(-0.48278\) \(\Gamma_0(N)\)-optimal
126.b6 126a3 \([1, -1, 1, 40, 155]\) \(9938375/21952\) \(-16003008\) \([6]\) \(24\) \(0.066527\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 126.2.a.b

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.