Properties

Label 2.256.acd_bwa
Base Field $\F_{2^{8}}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{8}}$
Dimension:  $2$
L-polynomial:  $( 1 - 16 x )^{2}( 1 - 23 x + 256 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.244714587078$
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.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 52650 4260438000 281429750633850 18446742823911948000 1208925273843386887466250 79228154008932884017451298000 5192296795851271534032910092221850 340282366605421791073310399807017368000 22300745197359433166677167781946632578340650 1461501637327805061502453133181808780658448750000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 202 65008 16774522 4294967008 1099511131402 281474946493648 72057593168018842 18446744056605358528 4722366482621636089642 1208925819612066687702448

Decomposition and endomorphism algebra

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

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.256.aj_aiq$2$(not in LMFDB)
2.256.j_aiq$2$(not in LMFDB)
2.256.cd_bwa$2$(not in LMFDB)
2.256.ah_fo$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.256.aj_aiq$2$(not in LMFDB)
2.256.j_aiq$2$(not in LMFDB)
2.256.cd_bwa$2$(not in LMFDB)
2.256.ah_fo$3$(not in LMFDB)
2.256.ax_ts$4$(not in LMFDB)
2.256.x_ts$4$(not in LMFDB)
2.256.abn_bhw$6$(not in LMFDB)
2.256.h_fo$6$(not in LMFDB)
2.256.bn_bhw$6$(not in LMFDB)