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

• Label: 3.9.ar_et_ass
• Base field: $\F_{3^{2}}$
• Dimension: $3$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3^{2}}$
Dimension:	$3$
L-polynomial:	$( 1 - 3 x )^{4}( 1 - 5 x + 9 x^{2} )$
	$1 - 17 x + 123 x^{2} - 486 x^{3} + 1107 x^{4} - 1377 x^{5} + 729 x^{6}$
Frobenius angles:	$0$, $0$, $0$, $0$, $\pm0.186429498677$
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:	$[0, 1/2, 1/2, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$80$	$307200$	$338162240$	$273408000000$	$204155398264400$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-7$	$39$	$632$	$6351$	$58553$	$529884$	$4776737$	$43020831$	$387320888$	$3486440679$

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

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

The isogeny class factors as 1.9.ag^2 $\times$ 1.9.af 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.ag^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 1.9.af : \(\Q(\sqrt{-11}) \).

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
3.9.ah_d_cc	$2$	(not in LMFDB)
3.9.af_aj_dm	$2$	(not in LMFDB)
3.9.f_aj_adm	$2$	(not in LMFDB)
3.9.h_d_acc	$2$	(not in LMFDB)
3.9.r_et_ss	$2$	(not in LMFDB)
3.9.ai_y_acc	$3$	(not in LMFDB)
3.9.b_g_abb	$3$	(not in LMFDB)