Properties

Label 159936.bb
Number of curves $6$
Conductor $159936$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
sage: E = EllipticCurve("bb1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 159936.bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
159936.bb1 159936in5 \([0, -1, 0, -43021542209, 3434621350524225]\) \(285531136548675601769470657/17941034271597192\) \(553319069389720922779287552\) \([2]\) \(283115520\) \(4.5918\)  
159936.bb2 159936in3 \([0, -1, 0, -2693962049, 53452178201409]\) \(70108386184777836280897/552468975892674624\) \(17038684335583337339676524544\) \([2, 2]\) \(141557760\) \(4.2452\)  
159936.bb3 159936in6 \([0, -1, 0, -917606209, 122887441088833]\) \(-2770540998624539614657/209924951154647363208\) \(-6474291105862282807622498254848\) \([2]\) \(283115520\) \(4.5918\)  
159936.bb4 159936in2 \([0, -1, 0, -284510529, -464118461631]\) \(82582985847542515777/44772582831427584\) \(1380830307296116429295714304\) \([2, 2]\) \(70778880\) \(3.8987\)  
159936.bb5 159936in1 \([0, -1, 0, -220285249, -1256774022335]\) \(38331145780597164097/55468445663232\) \(1710701193155990406561792\) \([2]\) \(35389440\) \(3.5521\) \(\Gamma_0(N)\)-optimal
159936.bb6 159936in4 \([0, -1, 0, 1097336511, -3651486844095]\) \(4738217997934888496063/2928751705237796928\) \(-90325571172963862651147911168\) \([2]\) \(141557760\) \(4.2452\)  

Rank

sage: E.rank()
 

The elliptic curves in class 159936.bb have rank \(0\).

Complex multiplication

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

Modular form 159936.2.a.bb

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2q^{5} + q^{9} - 4q^{11} - 2q^{13} + 2q^{15} - q^{17} + 4q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.