Properties

Label 27225.bx
Number of curves $3$
Conductor $27225$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bx1")
 
E.isogeny_class()
 

Elliptic curves in class 27225.bx

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
27225.bx1 27225bq3 \([0, 0, 1, -212908575, -1195742989719]\) \(-52893159101157376/11\) \(-221971057171875\) \([]\) \(2520000\) \(3.0497\)  
27225.bx2 27225bq2 \([0, 0, 1, -281325, -104025969]\) \(-122023936/161051\) \(-3249878248053421875\) \([]\) \(504000\) \(2.2450\)  
27225.bx3 27225bq1 \([0, 0, 1, -9075, 790281]\) \(-4096/11\) \(-221971057171875\) \([]\) \(100800\) \(1.4402\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 27225.bx do not have complex multiplication.

Modular form 27225.2.a.bx

sage: E.q_eigenform(10)
 
\(q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 4 q^{13} - 4 q^{14} - 4 q^{16} + 2 q^{17} + O(q^{20})\) 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.

\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.