Properties

Label 159936.dz
Number of curves $4$
Conductor $159936$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("dz1")
 
E.isogeny_class()
 

Elliptic curves in class 159936.dz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
159936.dz1 159936jz4 \([0, -1, 0, -249377, -47848863]\) \(444893916104/9639\) \(37159525122048\) \([2]\) \(688128\) \(1.7202\)  
159936.dz2 159936jz2 \([0, -1, 0, -16137, -687735]\) \(964430272/127449\) \(61416437354496\) \([2, 2]\) \(344064\) \(1.3737\)  
159936.dz3 159936jz1 \([0, -1, 0, -4132, 92590]\) \(1036433728/122451\) \(921999212736\) \([2]\) \(172032\) \(1.0271\) \(\Gamma_0(N)\)-optimal
159936.dz4 159936jz3 \([0, -1, 0, 25023, -3659487]\) \(449455096/1753941\) \(-6761657293504512\) \([4]\) \(688128\) \(1.7202\)  

Rank

sage: E.rank()
 

The elliptic curves in class 159936.dz have rank \(1\).

Complex multiplication

The elliptic curves in class 159936.dz do not have complex multiplication.

Modular form 159936.2.a.dz

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{5} + q^{9} - 2 q^{13} - 2 q^{15} + q^{17} + 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.