Properties

Label 90009b
Number of curves $2$
Conductor $90009$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("90009.b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 90009b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
90009.b2 90009b1 [1, -1, 1, -482348, -128819546] [2] 746496 \(\Gamma_0(N)\)-optimal
90009.b1 90009b2 [1, -1, 1, -485633, -126973376] [2] 1492992  

Rank

sage: E.rank()
 

The elliptic curves in class 90009b have rank \(1\).

Modular form 90009.2.a.b

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{4} + 4q^{5} - 4q^{7} + 3q^{8} - 4q^{10} - 4q^{11} + 4q^{14} - q^{16} - 2q^{17} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.