Properties

Label 27225.r
Number of curves $4$
Conductor $27225$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 27225.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
27225.r1 27225bo4 \([1, -1, 1, -3989030, 3064527722]\) \(347873904937/395307\) \(7976973881585671875\) \([2]\) \(737280\) \(2.5403\)  
27225.r2 27225bo2 \([1, -1, 1, -313655, 21317222]\) \(169112377/88209\) \(1779985907461265625\) \([2, 2]\) \(368640\) \(2.1938\)  
27225.r3 27225bo1 \([1, -1, 1, -177530, -28504528]\) \(30664297/297\) \(5993218543640625\) \([2]\) \(184320\) \(1.8472\) \(\Gamma_0(N)\)-optimal
27225.r4 27225bo3 \([1, -1, 1, 1183720, 165065222]\) \(9090072503/5845851\) \(-117964520594478421875\) \([2]\) \(737280\) \(2.5403\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 27225.2.a.r

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} + 4 q^{7} + 3 q^{8} - 2 q^{13} - 4 q^{14} - q^{16} + 2 q^{17} + O(q^{20})\) 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.