# Properties

 Label 89232bd Number of curves $6$ Conductor $89232$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("89232.z1")

sage: E.isogeny_class()

## Elliptic curves in class 89232bd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
89232.z5 89232bd1 [0, -1, 0, -64952, -9118032] [2] 688128 $$\Gamma_0(N)$$-optimal
89232.z4 89232bd2 [0, -1, 0, -1160072, -480457680] [2, 2] 1376256
89232.z3 89232bd3 [0, -1, 0, -1281752, -373379280] [2, 2] 2752512
89232.z1 89232bd4 [0, -1, 0, -18560312, -30770795472] [2] 2752512
89232.z6 89232bd5 [0, -1, 0, 3626008, -2548498512] [2] 5505024
89232.z2 89232bd6 [0, -1, 0, -8136392, 8652810672] [2] 5505024

## Rank

sage: E.rank()

The elliptic curves in class 89232bd have rank $$0$$.

## Modular form 89232.2.a.z

sage: E.q_eigenform(10)

$$q - q^{3} + 2q^{5} + q^{9} - q^{11} - 2q^{15} - 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.