Properties

Label 281775.bm
Number of curves $6$
Conductor $281775$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("281775.bm1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 281775.bm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
281775.bm1 281775bm6 [1, 1, 0, -145757300, -677276009625] [2] 37748736  
281775.bm2 281775bm4 [1, 1, 0, -10035675, -8304120000] [2, 2] 18874368  
281775.bm3 281775bm2 [1, 1, 0, -3930550, 2898784375] [2, 2] 9437184  
281775.bm4 281775bm1 [1, 1, 0, -3894425, 2956476000] [2] 4718592 \(\Gamma_0(N)\)-optimal
281775.bm5 281775bm3 [1, 1, 0, 1596575, 10410147250] [2] 18874368  
281775.bm6 281775bm5 [1, 1, 0, 28003950, -56196007875] [2] 37748736  

Rank

sage: E.rank()
 

The elliptic curves in class 281775.bm have rank \(1\).

Modular form 281775.2.a.bm

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{3} - q^{4} - q^{6} - 3q^{8} + q^{9} - 4q^{11} + q^{12} - q^{13} - q^{16} + q^{18} - 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.