Properties

Label 27225.bp
Number of curves 8
Conductor 27225
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("27225.bp1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 27225.bp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
27225.bp1 27225bi8 [1, -1, 0, -58806567, -173560171284] [2] 983040  
27225.bp2 27225bi6 [1, -1, 0, -3675942, -2710364409] [2, 2] 491520  
27225.bp3 27225bi7 [1, -1, 0, -2995317, -3745595034] [2] 983040  
27225.bp4 27225bi4 [1, -1, 0, -2178567, 1238213466] [2] 245760  
27225.bp5 27225bi3 [1, -1, 0, -272817, -25298784] [2, 2] 245760  
27225.bp6 27225bi2 [1, -1, 0, -136692, 19214091] [2, 2] 122880  
27225.bp7 27225bi1 [1, -1, 0, -567, 837216] [2] 61440 \(\Gamma_0(N)\)-optimal
27225.bp8 27225bi5 [1, -1, 0, 952308, -190690659] [2] 491520  

Rank

sage: E.rank()
 

The elliptic curves in class 27225.bp have rank \(1\).

Modular form 27225.2.a.bp

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

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 LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.