Properties

Label 20535a
Number of curves 8
Conductor 20535
CM no
Rank 1
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath

sage: E = EllipticCurve("20535.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 20535a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
20535.f7 20535a1 [1, 1, 0, -28, 9427] [2] 12096 \(\Gamma_0(N)\)-optimal
20535.f6 20535a2 [1, 1, 0, -6873, 213408] [2, 2] 24192  
20535.f5 20535a3 [1, 1, 0, -13718, -291753] [2, 2] 48384  
20535.f4 20535a4 [1, 1, 0, -109548, 13910253] [2] 48384  
20535.f2 20535a5 [1, 1, 0, -184843, -30649328] [2, 2] 96768  
20535.f8 20535a6 [1, 1, 0, 47887, -2127582] [2] 96768  
20535.f1 20535a7 [1, 1, 0, -2957068, -1958454593] [2] 193536  
20535.f3 20535a8 [1, 1, 0, -150618, -42306363] [2] 193536  

Rank

sage: E.rank()
 

The elliptic curves in class 20535a have rank \(1\).

Modular form 20535.2.a.f

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.