Properties

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

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Show commands for: SageMath

sage: E = EllipticCurve("53361.p1")
sage: E.isogeny_class()

Elliptic curves in class 53361bo

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
53361.p6 53361bo1 [1, -1, 1, 52249, 999870] 2 245760 \(\Gamma_0(N)\)-optimal
53361.p5 53361bo2 [1, -1, 1, -214556, 8256966] 4 491520  
53361.p3 53361bo3 [1, -1, 1, -2082191, -1148929680] 2 983040  
53361.p2 53361bo4 [1, -1, 1, -2615801, 1626696096] 4 983040  
53361.p4 53361bo5 [1, -1, 1, -1815386, 2640341652] 2 1966080  
53361.p1 53361bo6 [1, -1, 1, -41836136, 104164339920] 2 1966080  

Rank

sage: E.rank()

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

Modular form 53361.2.a.p

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

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.