Properties

Label 2.1024.aey_jci
Base Field $\F_{2^{10}}$
Dimension $2$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{10}}$
Dimension:  $2$
L-polynomial:  $( 1 - 32 x )^{4}$
Frobenius angles:  $0$, $0$, $0$, $0$
Angle rank:  $0$ (numerical)

This isogeny class is not simple.

Newton polygon

This isogeny class is supersingular.

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

Point counts

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

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 923521 1095222947841 1152780773560811521 1208921207935207812890625 1267650449112508705067363205121 1329227990833155722679814984029962241 1393796574745904669523852578373111612702721 1461501637325586006220552422779585474791812890625 1532495540865714633786483514084656171531874968765726721 1606938044258984566551191268509244272979694567707627896176641

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 897 1044481 1073610753 1099507433473 1125899772624897 1152921500311879681 1180591620579972349953 1208925819610231128195073 1237940039285239537410768897 1267650600228224897897075834881

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
The isogeny class factors as 1.1024.acm 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{10}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{10}}$.

SubfieldPrimitive Model
$\F_{2}$2.2.a_ae

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.1024.a_adau$2$(not in LMFDB)
2.1024.ey_jci$2$(not in LMFDB)
2.1024.abg_a$3$(not in LMFDB)
2.1024.cm_eoe$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.1024.a_adau$2$(not in LMFDB)
2.1024.ey_jci$2$(not in LMFDB)
2.1024.abg_a$3$(not in LMFDB)
2.1024.cm_eoe$3$(not in LMFDB)
2.1024.acm_dau$4$(not in LMFDB)
2.1024.a_dau$4$(not in LMFDB)
2.1024.cm_dau$4$(not in LMFDB)
2.1024.bg_bnk$5$(not in LMFDB)
2.1024.ads_gbo$6$(not in LMFDB)
2.1024.acm_eoe$6$(not in LMFDB)
2.1024.a_bnk$6$(not in LMFDB)
2.1024.bg_a$6$(not in LMFDB)
2.1024.ds_gbo$6$(not in LMFDB)
2.1024.a_a$8$(not in LMFDB)
2.1024.abg_bnk$10$(not in LMFDB)
2.1024.abg_dau$12$(not in LMFDB)
2.1024.a_abnk$12$(not in LMFDB)
2.1024.bg_dau$12$(not in LMFDB)