Properties

Label 15225.w
Number of curves $6$
Conductor $15225$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("15225.w1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 15225.w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
15225.w1 15225o4 [1, 0, 1, -5115601, 4452987473] [2] 196608  
15225.w2 15225o5 [1, 0, 1, -1061726, -342394027] [2] 393216  
15225.w3 15225o3 [1, 0, 1, -325851, 66752473] [2, 2] 196608  
15225.w4 15225o2 [1, 0, 1, -319726, 69557723] [2, 2] 98304  
15225.w5 15225o1 [1, 0, 1, -19601, 1129223] [2] 49152 \(\Gamma_0(N)\)-optimal
15225.w6 15225o6 [1, 0, 1, 312024, 296387473] [2] 393216  

Rank

sage: E.rank()
 

The elliptic curves in class 15225.w have rank \(1\).

Modular form 15225.2.a.w

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} - q^{4} + q^{6} - q^{7} - 3q^{8} + q^{9} + 4q^{11} - q^{12} + 2q^{13} - q^{14} - q^{16} - 2q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.