Properties

Label 136242.z
Number of curves $2$
Conductor $136242$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("z1")
 
E.isogeny_class()
 

Elliptic curves in class 136242.z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
136242.z1 136242i1 \([1, -1, 1, -5204, 159067]\) \(-35937/4\) \(-1734504804036\) \([]\) \(275184\) \(1.0858\) \(\Gamma_0(N)\)-optimal
136242.z2 136242i2 \([1, -1, 1, 32641, -234521]\) \(109503/64\) \(-2247918226030656\) \([]\) \(825552\) \(1.6351\)  

Rank

sage: E.rank()
 

The elliptic curves in class 136242.z have rank \(0\).

Complex multiplication

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

Modular form 136242.2.a.z

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.