Abelian variety isogeny class 4.3.h_bb_ct_fk over $\F_{3}$

• Label: 4.3.h_bb_ct_fk
• Base field: $\F_{3}$
• Dimension: $4$
• $p$-rank: $4$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3}$
Dimension:	$4$
L-polynomial:	$( 1 + 2 x + 3 x^{2} )^{2}( 1 + 3 x + 5 x^{2} + 9 x^{3} + 9 x^{4} )$
	$1 + 7 x + 27 x^{2} + 71 x^{3} + 140 x^{4} + 213 x^{5} + 243 x^{6} + 189 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.472142961319$, $\pm0.695913276015$, $\pm0.695913276015$, $\pm0.902473643995$
Angle rank:	$3$ (numerical)
Jacobians:	$1$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$4$
Slopes:	$[0, 0, 0, 0, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$972$	$11664$	$341172$	$45536256$	$3424721472$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$11$	$15$	$17$	$87$	$236$	$699$	$2363$	$6495$	$18845$	$60810$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is not hyperelliptic):

• $x t-y z=x y^2+x^2 z-z^3+x^2 t+x y t+y^2 t+x t^2-y t^2+z t^2=0$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.c^2 $\times$ 2.3.d_f and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.3.c^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
• 2.3.d_f : 4.0.2197.1.

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
4.3.ah_bb_act_fk	$2$	(not in LMFDB)
4.3.ad_h_ap_bc	$2$	(not in LMFDB)
4.3.ab_d_h_ae	$2$	(not in LMFDB)
4.3.b_d_ah_ae	$2$	(not in LMFDB)
4.3.d_h_p_bc	$2$	(not in LMFDB)
4.3.b_a_ae_an	$3$	(not in LMFDB)