# Properties

 Label 89232u Number of curves 4 Conductor 89232 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("89232.bo1")

sage: E.isogeny_class()

## Elliptic curves in class 89232u

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
89232.bo3 89232u1 [0, 1, 0, -2084, 34620]  73728 $$\Gamma_0(N)$$-optimal
89232.bo2 89232u2 [0, 1, 0, -5464, -110044] [2, 2] 147456
89232.bo4 89232u3 [0, 1, 0, 14816, -718444]  294912
89232.bo1 89232u4 [0, 1, 0, -79824, -8706060]  294912

## Rank

sage: E.rank()

The elliptic curves in class 89232u have rank $$1$$.

## Modular form 89232.2.a.bo

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{5} + q^{9} + q^{11} - 2q^{15} + 6q^{17} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 