Abelian variety isogeny class 3.9.ad_p_acg over $\F_{3^{2}}$

• Label: 3.9.ad_p_acg
• Base field: $\F_{3^{2}}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3^{2}}$
Dimension:	$3$
L-polynomial:	$( 1 + 2 x + 9 x^{2} )( 1 - 5 x + 16 x^{2} - 45 x^{3} + 81 x^{4} )$
	$1 - 3 x + 15 x^{2} - 58 x^{3} + 135 x^{4} - 243 x^{5} + 729 x^{6}$
Frobenius angles:	$\pm0.146903834656$, $\pm0.519762832011$, $\pm0.608173447969$
Angle rank:	$2$ (numerical)
Isomorphism classes:	1762

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$576$	$681984$	$354585600$	$276208975872$	$209254963807296$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$7$	$103$	$664$	$6415$	$60007$	$533500$	$4782463$	$43058335$	$387465496$	$3486777703$

Jacobians and polarizations

This isogeny class contains a Jacobian, and hence is principally polarizable.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{6}}$.

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

The isogeny class factors as 1.9.c $\times$ 2.9.af_q and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.9.c : \(\Q(\sqrt{-2}) \).
• 2.9.af_q : \(\Q(\sqrt{-3}, \sqrt{-11})\).

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.ak^2. The endomorphism algebra for each factor is:

• 1.729.abu : \(\Q(\sqrt{-2}) \).
• 1.729.ak^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{2}}$.

Subfield	Primitive Model
$\F_{3}$	3.3.ad_d_ac
$\F_{3}$	3.3.ab_ab_k
$\F_{3}$	3.3.b_ab_ak
$\F_{3}$	3.3.d_d_c

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
3.9.ah_bj_aes	$2$	(not in LMFDB)
3.9.d_p_cg	$2$	(not in LMFDB)
3.9.h_bj_es	$2$	(not in LMFDB)
3.9.m_cu_kg	$3$	(not in LMFDB)