Properties

Label 441525.k
Number of curves $6$
Conductor $441525$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("441525.k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 441525.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
441525.k1 441525k3 [1, 1, 1, -4302220038, 108612515925156] [2] 165150720 \(\Gamma_0(N)\)-optimal*
441525.k2 441525k6 [1, 1, 1, -892911163, -8348862096094] [2] 330301440  
441525.k3 441525k4 [1, 1, 1, -274040288, 1628574150656] [2, 2] 165150720  
441525.k4 441525k2 [1, 1, 1, -268889163, 1696981090656] [2, 2] 82575360 \(\Gamma_0(N)\)-optimal*
441525.k5 441525k1 [1, 1, 1, -16484038, 27573593906] [2] 41287680 \(\Gamma_0(N)\)-optimal*
441525.k6 441525k5 [1, 1, 1, 262412587, 7228069259906] [2] 330301440  
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 441525.k5.

Rank

sage: E.rank()
 

The elliptic curves in class 441525.k have rank \(1\).

Modular form 441525.2.a.k

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.