Abelian variety isogeny class 2.8.ae_i over $\F_{2^{3}}$

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

Invariants

Base field:	$\F_{2^{3}}$
Dimension:	$2$
L-polynomial:	$1 - 4 x + 8 x^{2} - 32 x^{3} + 64 x^{4}$
Frobenius angles:	$\pm0.0833333333333$, $\pm0.583333333333$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\zeta_{12})\)
Galois group:	$C_2^2$
Jacobians:	$3$

This isogeny class is simple but not geometrically simple, 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$37$	$4033$	$231361$	$16265089$	$1082163457$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$65$	$449$	$3969$	$33025$	$262145$	$2095105$	$16785409$	$134250497$	$1073741825$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2+y=(a^2+a+1)x^5+ax^4+(a^2+a+1)x^3+1$
• $y^2+y=(a^2+1)x^5+a^2x^4+(a^2+1)x^3+1$
• $y^2+y=(a+1)x^5+a^2x^4+(a+1)x^3+1$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{36}}$.

Endomorphism algebra over $\F_{2^{3}}$

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{12})\).

Endomorphism algebra over $\overline{\F}_{2^{3}}$

The base change of $A$ to $\F_{2^{36}}$ is 1.68719476736.bdvoy^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

• Endomorphism algebra over $\F_{2^{6}}$

  The base change of $A$ to $\F_{2^{6}}$ is the simple isogeny class 2.64.a_acm and its endomorphism algebra is \(\Q(\zeta_{12})\).

• Endomorphism algebra over $\F_{2^{9}}$

  The base change of $A$ to $\F_{2^{9}}$ is 1.512.abg^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

• Endomorphism algebra over $\F_{2^{12}}$

  The base change of $A$ to $\F_{2^{12}}$ is 1.4096.acm^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

• Endomorphism algebra over $\F_{2^{18}}$

  The base change of $A$ to $\F_{2^{18}}$ is 1.262144.a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.8.e_i	$2$	2.64.a_acm
2.8.i_bg	$3$	2.512.acm_dau
2.8.ai_bg	$6$	(not in LMFDB)