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

• Label: 3.9.ap_dx_apa
• Base field: $\F_{3^{2}}$
• Dimension: $3$
• $p$-rank: $2$
• 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:	$( 1 - 3 x )^{2}( 1 - 5 x + 9 x^{2} )( 1 - 4 x + 9 x^{2} )$
	$1 - 15 x + 101 x^{2} - 390 x^{3} + 909 x^{4} - 1215 x^{5} + 729 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.186429498677$, $\pm0.267720472801$
Angle rank:	$2$ (numerical)

This isogeny class is not simple, 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$120$	$403200$	$387185760$	$287078400000$	$206672306220600$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$59$	$730$	$6671$	$59275$	$530864$	$4777075$	$43022111$	$387349210$	$3486621179$

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 $\times$ 1.9.af $\times$ 1.9.ae 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 : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 1.9.af : \(\Q(\sqrt{-11}) \).
• 1.9.ae : \(\Q(\sqrt{-5}) \).

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_n_ag	$2$	(not in LMFDB)
3.9.af_b_be	$2$	(not in LMFDB)
3.9.ad_ah_co	$2$	(not in LMFDB)
3.9.d_ah_aco	$2$	(not in LMFDB)
3.9.f_b_abe	$2$	(not in LMFDB)
3.9.h_n_g	$2$	(not in LMFDB)
3.9.p_dx_pa	$2$	(not in LMFDB)
3.9.ag_u_abw	$3$	(not in LMFDB)