Properties

Label 2.8.ai_bg
Base field $\F_{2^{3}}$
Dimension $2$
$p$-rank $0$
Ordinary no
Supersingular yes
Simple no
Geometrically simple no
Primitive no
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{2^{3}}$
Dimension:  $2$
L-polynomial:  $( 1 - 4 x + 8 x^{2} )^{2}$
  $1 - 8 x + 32 x^{2} - 64 x^{3} + 64 x^{4}$
Frobenius angles:  $\pm0.250000000000$, $\pm0.250000000000$
Angle rank:  $0$ (numerical)
Jacobians:  $1$

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $25$ $4225$ $297025$ $17850625$ $1090650625$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $65$ $577$ $4353$ $33281$ $262145$ $2093057$ $16760833$ $134184961$ $1073741825$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{12}}$.

Endomorphism algebra over $\F_{2^{3}}$
The isogeny class factors as 1.8.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
Endomorphism algebra over $\overline{\F}_{2^{3}}$
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
Remainder of endomorphism lattice by field

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}$2.2.ac_c
$\F_{2}$2.2.e_i

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.8.a_a$2$2.64.a_ey
2.8.i_bg$2$2.64.a_ey
2.8.e_i$3$2.512.cm_dau

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

TwistExtension degreeCommon base change
2.8.a_a$2$2.64.a_ey
2.8.i_bg$2$2.64.a_ey
2.8.e_i$3$2.512.cm_dau
2.8.ae_i$6$(not in LMFDB)
2.8.ae_q$8$(not in LMFDB)
2.8.a_aq$8$(not in LMFDB)
2.8.a_q$8$(not in LMFDB)
2.8.e_q$8$(not in LMFDB)
2.8.a_ai$24$(not in LMFDB)
2.8.a_i$24$(not in LMFDB)