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

Learn more about

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.

Point counts of the abelian variety

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

Point counts of the (virtual) curve

$r$ 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:
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.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)