Properties

Label 2.256.acm_chc
Base Field $\F_{2^{8}}$
Dimension $2$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{8}}$
Dimension:  $2$
L-polynomial:  $( 1 - 16 x )^{4}$
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
$A(\F_{q^r})$ 50625 4228250625 281200199450625 18445618199572250625 1208921207935207812890625 79228143624800094964756250625 5192296781163575605539792301850625 340282366604025813516997721482669850625 22300745197232548926930345564412680417050625 1461501637325586006220552422779585474791812890625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 193 64513 16760833 4294705153 1099507433473 281474909601793 72057592964186113 18446744056529682433 4722366482594767306753 1208925819610231128195073

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{8}}$
The isogeny class factors as 1.256.abg 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 $\F_{2^{8}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{8}}$.

SubfieldPrimitive Model
$\F_{2}$2.2.ae_i
$\F_{2}$2.2.ac_e
$\F_{2}$2.2.a_ae
$\F_{2}$2.2.a_a
$\F_{2}$2.2.a_e
$\F_{2}$2.2.c_e
$\F_{2}$2.2.e_i

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.256.a_ats$2$(not in LMFDB)
2.256.cm_chc$2$(not in LMFDB)
2.256.aq_a$3$(not in LMFDB)
2.256.bg_bdo$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.256.a_ats$2$(not in LMFDB)
2.256.cm_chc$2$(not in LMFDB)
2.256.aq_a$3$(not in LMFDB)
2.256.bg_bdo$3$(not in LMFDB)
2.256.abg_ts$4$(not in LMFDB)
2.256.a_ts$4$(not in LMFDB)
2.256.bg_ts$4$(not in LMFDB)
2.256.q_jw$5$(not in LMFDB)
2.256.abw_bnk$6$(not in LMFDB)
2.256.abg_bdo$6$(not in LMFDB)
2.256.a_jw$6$(not in LMFDB)
2.256.q_a$6$(not in LMFDB)
2.256.bw_bnk$6$(not in LMFDB)
2.256.a_a$8$(not in LMFDB)
2.256.aq_jw$10$(not in LMFDB)
2.256.aq_ts$12$(not in LMFDB)
2.256.a_ajw$12$(not in LMFDB)
2.256.q_ts$12$(not in LMFDB)