Properties

Label 2005.b
Number of curves 4
Conductor 2005
CM no
Rank 1
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("2005.b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 2005.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2005.b1 2005a3 [1, -1, 1, -53467, -4745166] [2] 1792  
2005.b2 2005a2 [1, -1, 1, -3342, -73516] [2, 2] 896  
2005.b3 2005a4 [1, -1, 1, -3217, -79366] [4] 1792  
2005.b4 2005a1 [1, -1, 1, -217, -1016] [4] 448 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 2005.b have rank \(1\).

Modular form 2005.2.a.b

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.