Properties

Label 53361bp
Number of curves $6$
Conductor $53361$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bp1")
 
E.isogeny_class()
 

Elliptic curves in class 53361bp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
53361.q4 53361bp1 \([1, -1, 1, -1815386, -940703200]\) \(4354703137/1617\) \(245686842692252577\) \([2]\) \(921600\) \(2.3043\) \(\Gamma_0(N)\)-optimal
53361.q3 53361bp2 \([1, -1, 1, -2082191, -645830314]\) \(6570725617/2614689\) \(397275624633372417009\) \([2, 2]\) \(1843200\) \(2.6509\)  
53361.q6 53361bp3 \([1, -1, 1, 6722374, -4681842910]\) \(221115865823/190238433\) \(-28904811355900823431473\) \([2]\) \(3686400\) \(2.9974\)  
53361.q2 53361bp4 \([1, -1, 1, -15155636, 22258845326]\) \(2533811507137/58110129\) \(8829248065831480859649\) \([2, 2]\) \(3686400\) \(2.9974\)  
53361.q5 53361bp5 \([1, -1, 1, 1653079, 68913114680]\) \(3288008303/13504609503\) \(-2051889221140297004224143\) \([2]\) \(7372800\) \(3.3440\)  
53361.q1 53361bp6 \([1, -1, 1, -241139471, 1441346935592]\) \(10206027697760497/5557167\) \(844355482092502892127\) \([2]\) \(7372800\) \(3.3440\)  

Rank

sage: E.rank()
 

The elliptic curves in class 53361bp have rank \(0\).

Complex multiplication

The elliptic curves in class 53361bp do not have complex multiplication.

Modular form 53361.2.a.bp

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} - 2 q^{5} + 3 q^{8} + 2 q^{10} + 6 q^{13} - q^{16} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.