Properties

Label 15.a
Number of curves 8
Conductor 15
CM no
Rank 0
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath

sage: E = EllipticCurve("15.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 15.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
15.a1 15a5 [1, 1, 1, -2160, -39540] [2] 4  
15.a2 15a2 [1, 1, 1, -135, -660] [2, 2] 2  
15.a3 15a6 [1, 1, 1, -110, -880] [2] 4  
15.a4 15a7 [1, 1, 1, -80, 242] [4] 4  
15.a5 15a1 [1, 1, 1, -10, -10] [2, 4] 1 \(\Gamma_0(N)\)-optimal
15.a6 15a3 [1, 1, 1, -5, 2] [2, 4] 2  
15.a7 15a8 [1, 1, 1, 0, 0] [4] 4  
15.a8 15a4 [1, 1, 1, 35, -28] [8] 2  

Rank

sage: E.rank()
 

The elliptic curves in class 15.a have rank \(0\).

Modular form 15.2.a.a

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + 3q^{8} + q^{9} - q^{10} - 4q^{11} + q^{12} - 2q^{13} - q^{15} - q^{16} + 2q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.