Properties

Label 3.8.am_cp_aiy
Base field $\F_{2^{3}}$
Dimension $3$
$p$-rank $2$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive no
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{2^{3}}$
Dimension:  $3$
L-polynomial:  $( 1 - 5 x + 8 x^{2} )( 1 - 7 x + 24 x^{2} - 56 x^{3} + 64 x^{4} )$
  $1 - 12 x + 67 x^{2} - 232 x^{3} + 536 x^{4} - 768 x^{5} + 512 x^{6}$
Frobenius angles:  $\pm0.0585111942353$, $\pm0.154919815756$, $\pm0.418160225599$
Angle rank:  $3$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $104$ $221312$ $131247896$ $66851273216$ $34871453017544$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-3$ $55$ $501$ $3983$ $32477$ $262711$ $2101957$ $16786207$ $134210157$ $1073693895$

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 $\F_{2^{3}}$.

Endomorphism algebra over $\F_{2^{3}}$
The isogeny class factors as 1.8.af $\times$ 2.8.ah_y 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 isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{3}}$.

SubfieldPrimitive Model
$\F_{2}$3.2.a_b_ae

Twists

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

TwistExtension degreeCommon base change
3.8.ac_ad_i$2$(not in LMFDB)
3.8.c_ad_ai$2$(not in LMFDB)
3.8.m_cp_iy$2$(not in LMFDB)