Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("bc1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 159936.bc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
159936.bc1 159936cz4 \([0, -1, 0, -106689, -12154911]\) \(17418812548/1753941\) \(13523314587009024\) \([2]\) \(983040\) \(1.8314\)  
159936.bc2 159936cz2 \([0, -1, 0, -24369, 1263249]\) \(830321872/127449\) \(245665749417984\) \([2, 2]\) \(491520\) \(1.4848\)  
159936.bc3 159936cz1 \([0, -1, 0, -23389, 1384573]\) \(11745974272/357\) \(43008709632\) \([2]\) \(245760\) \(1.1383\) \(\Gamma_0(N)\)-optimal
159936.bc4 159936cz3 \([0, -1, 0, 42271, 6900993]\) \(1083360092/3306177\) \(-25491434233724928\) \([2]\) \(983040\) \(1.8314\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 159936.2.a.bc

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2q^{5} + q^{9} - 4q^{11} + 6q^{13} + 2q^{15} - q^{17} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.