Properties

Label 2.256.acj_cdk
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:  $( 1 - 16 x )^{2}( 1 - 29 x + 256 x^{2} )$
  $1 - 61 x + 1440 x^{2} - 15616 x^{3} + 65536 x^{4}$
Frobenius angles:  $0$, $0$, $\pm0.138932406859$
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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $51300$ $4240150200$ $281302071277500$ $18446279187751786800$ $1208924837639368840732500$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $196$ $64696$ $16766908$ $4294859056$ $1099510734676$ $281474972228968$ $72057594035929516$ $18446744073188297056$ $4722366482829270623428$ $1208925819613179435895576$

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.abd 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.ad_aqa$2$(not in LMFDB)
2.256.d_aqa$2$(not in LMFDB)
2.256.cj_cdk$2$(not in LMFDB)
2.256.an_bw$3$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.256.ad_aqa$2$(not in LMFDB)
2.256.d_aqa$2$(not in LMFDB)
2.256.cj_cdk$2$(not in LMFDB)
2.256.an_bw$3$(not in LMFDB)
2.256.abd_ts$4$(not in LMFDB)
2.256.bd_ts$4$(not in LMFDB)
2.256.abt_blo$6$(not in LMFDB)
2.256.n_bw$6$(not in LMFDB)
2.256.bt_blo$6$(not in LMFDB)