# Properties

 Label 2.1024.aep_ige Base Field $\F_{2^{10}}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{10}}$ Dimension: $2$ L-polynomial: $( 1 - 32 x )^{2}( 1 - 55 x + 1024 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.170852887823$ Angle rank: $1$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1]$

## Point counts

This isogeny class contains a Jacobian, and hence is principally polarizable.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 932170 1096343780400 1152853913463294490 1208924770098697393500000 1267650592446413907299277375850 1329227995777211789711192531945523600 1393796574893268000546984411502451776186810 1461501637329324559982619867313794171301971000000 1532495540865791459292929135192228270552006218300619530 1606938044258985681949005821574657304299783129639078675510000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 906 1045552 1073678874 1099510673248 1125899899930986 1152921504600164752 1180591620704793946554 1208925819613323587386048 1237940039285301596560653066 1267650600228225777790805504752

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
 The isogeny class factors as 1.1024.acm $\times$ 1.1024.acd and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.1024.acm : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.1024.acd : $$\Q(\sqrt{-119})$$.
All geometric endomorphisms are defined over $\F_{2^{10}}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.1024.aj_aceq $2$ (not in LMFDB) 2.1024.j_aceq $2$ (not in LMFDB) 2.1024.ep_ige $2$ (not in LMFDB) 2.1024.ax_lc $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.1024.aj_aceq $2$ (not in LMFDB) 2.1024.j_aceq $2$ (not in LMFDB) 2.1024.ep_ige $2$ (not in LMFDB) 2.1024.ax_lc $3$ (not in LMFDB) 2.1024.acd_dau $4$ (not in LMFDB) 2.1024.cd_dau $4$ (not in LMFDB) 2.1024.adj_fqm $6$ (not in LMFDB) 2.1024.x_lc $6$ (not in LMFDB) 2.1024.dj_fqm $6$ (not in LMFDB)