Properties

Label 136242.bj
Number of curves $4$
Conductor $136242$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bj1")
 
E.isogeny_class()
 

Elliptic curves in class 136242.bj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
136242.bj1 136242e3 \([1, -1, 1, -905915, 332105131]\) \(-189613868625/128\) \(-55504153729152\) \([]\) \(1016064\) \(1.9530\)  
136242.bj2 136242e4 \([1, -1, 1, -716690, 474568849]\) \(-1159088625/2097152\) \(-73659784430572535808\) \([]\) \(3048192\) \(2.5023\)  
136242.bj3 136242e2 \([1, -1, 1, -35480, -2686877]\) \(-140625/8\) \(-280989778253832\) \([]\) \(435456\) \(1.5294\)  
136242.bj4 136242e1 \([1, -1, 1, 2365, -7451]\) \(3375/2\) \(-867252402018\) \([]\) \(145152\) \(0.98006\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 136242.bj have rank \(1\).

Complex multiplication

The elliptic curves in class 136242.bj do not have complex multiplication.

Modular form 136242.2.a.bj

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

\(\left(\begin{array}{rrrr} 1 & 3 & 21 & 7 \\ 3 & 1 & 7 & 21 \\ 21 & 7 & 1 & 3 \\ 7 & 21 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.