Properties

Label 2.256.ach_cay
Base field $\F_{2^{8}}$
Dimension $2$
$p$-rank $1$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{2^{8}}$
Dimension:  $2$
L-polynomial:  $( 1 - 16 x )^{2}( 1 - 27 x + 256 x^{2} )$
  $1 - 59 x + 1376 x^{2} - 15104 x^{3} + 65536 x^{4}$
Frobenius angles:  $0$, $0$, $\pm0.180343027596$
Angle rank:  $1$ (numerical)

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $51750$ $4247433000$ $281355229086750$ $18446541825576618000$ $1208925710564174413593750$

Point counts of the (virtual) curve

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

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{8}}$.

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:

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)