# Properties

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

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]  746496 $$\Gamma_0(N)$$-optimal
90009.b1 90009b2 [1, -1, 1, -485633, -126973376]  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. 