Properties

Label 53361.bm
Number of curves $4$
Conductor $53361$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bm1")
 
E.isogeny_class()
 

Elliptic curves in class 53361.bm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
53361.bm1 53361bj4 \([1, -1, 0, -7818498, -8404372971]\) \(347873904937/395307\) \(60062912012459053467\) \([2]\) \(1658880\) \(2.7086\)  
53361.bm2 53361bj2 \([1, -1, 0, -614763, -58125600]\) \(169112377/88209\) \(13402467969722268129\) \([2, 2]\) \(829440\) \(2.3620\)  
53361.bm3 53361bj1 \([1, -1, 0, -347958, 78425199]\) \(30664297/297\) \(45126154780209657\) \([2]\) \(414720\) \(2.0154\) \(\Gamma_0(N)\)-optimal
53361.bm4 53361bj3 \([1, -1, 0, 2320092, -454331025]\) \(9090072503/5845851\) \(-888218104538866678731\) \([2]\) \(1658880\) \(2.7086\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 53361.2.a.bm

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