Properties

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

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Invariants

Base field:  $\F_{2^{8}}$
Dimension:  $2$
L-polynomial:  $( 1 - 16 x )^{2}( 1 - 27 x + 256 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.180343027596$
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
$A(\F_{q^r})$ 51750 4247433000 281355229086750 18446541825576618000 1208925710564174413593750 79228162202161824397479843000 5192296846117850074003925016156750 340282366790830915872938048362603188000 22300745197636817990040624128736126131751750 1461501637326077011928629537952148728325508125000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 198 64808 16770078 4294920208 1099511528598 281474975601848 72057593865607758 18446744066656407328 4722366482680374610758 1208925819610637278597448

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{8}}$
The isogeny class factors as 1.256.abg $\times$ 1.256.abb 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.af_ano$2$(not in LMFDB)
2.256.f_ano$2$(not in LMFDB)
2.256.ch_cay$2$(not in LMFDB)
2.256.al_dc$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.256.af_ano$2$(not in LMFDB)
2.256.f_ano$2$(not in LMFDB)
2.256.ch_cay$2$(not in LMFDB)
2.256.al_dc$3$(not in LMFDB)
2.256.abb_ts$4$(not in LMFDB)
2.256.bb_ts$4$(not in LMFDB)
2.256.abr_bki$6$(not in LMFDB)
2.256.l_dc$6$(not in LMFDB)
2.256.br_bki$6$(not in LMFDB)