Properties

Label 53550.du
Number of curves $6$
Conductor $53550$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("du1")
 
E.isogeny_class()
 

Elliptic curves in class 53550.du

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
53550.du1 53550eb6 \([1, -1, 1, -392080505, 2988303444497]\) \(585196747116290735872321/836876053125000\) \(9532541292626953125000\) \([2]\) \(14155776\) \(3.4887\)  
53550.du2 53550eb4 \([1, -1, 1, -56839505, -164898629503]\) \(1782900110862842086081/328139630024640\) \(3737715473249415000000\) \([2]\) \(7077888\) \(3.1421\)  
53550.du3 53550eb3 \([1, -1, 1, -24727505, 45805914497]\) \(146796951366228945601/5397929064360000\) \(61485785748725625000000\) \([2, 2]\) \(7077888\) \(3.1421\)  
53550.du4 53550eb2 \([1, -1, 1, -3919505, -2010869503]\) \(584614687782041281/184812061593600\) \(2105124889089600000000\) \([2, 2]\) \(3538944\) \(2.7955\)  
53550.du5 53550eb1 \([1, -1, 1, 688495, -213749503]\) \(3168685387909439/3563732336640\) \(-40593138647040000000\) \([4]\) \(1769472\) \(2.4490\) \(\Gamma_0(N)\)-optimal
53550.du6 53550eb5 \([1, -1, 1, 9697495, 163332864497]\) \(8854313460877886399/1016927675429790600\) \(-11583441802942458553125000\) \([2]\) \(14155776\) \(3.4887\)  

Rank

sage: E.rank()
 

The elliptic curves in class 53550.du have rank \(1\).

Complex multiplication

The elliptic curves in class 53550.du do not have complex multiplication.

Modular form 53550.2.a.du

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{7} + q^{8} - 4 q^{11} - 6 q^{13} + q^{14} + q^{16} + q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.