# Properties

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

## Invariants

 Base field: $\F_{2^{10}}$ Dimension: $2$ L-polynomial: $( 1 - 32 x )^{2}( 1 - 63 x + 1024 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.0563432964760$ 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 is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 924482 1095355857024 1152790463489319746 1208921762157280348320000 1267650476822616254571456465602 1329227992103068983441002677865490816 1393796574800611111955059327269855856964162 1461501637327834746001140995303076039656796480000 1532495540865803686150860416321292237895023691212855746 1606938044258987986134349030743589871334775685150679544715904

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 898 1044608 1073619778 1099507937536 1125899797236418 1152921501413353856 1180591620626310509122 1208925819612091242144256 1237940039285311473337833922 1267650600228227595472541145728

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
 The isogeny class factors as 1.1024.acm $\times$ 1.1024.acl 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.acl : $$\Q(\sqrt{-127})$$.
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.ab_acyi $2$ (not in LMFDB) 2.1024.b_acyi $2$ (not in LMFDB) 2.1024.ex_izw $2$ (not in LMFDB) 2.1024.abf_bg $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.1024.ab_acyi $2$ (not in LMFDB) 2.1024.b_acyi $2$ (not in LMFDB) 2.1024.ex_izw $2$ (not in LMFDB) 2.1024.abf_bg $3$ (not in LMFDB) 2.1024.acl_dau $4$ (not in LMFDB) 2.1024.cl_dau $4$ (not in LMFDB) 2.1024.adr_gai $6$ (not in LMFDB) 2.1024.bf_bg $6$ (not in LMFDB) 2.1024.dr_gai $6$ (not in LMFDB)