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

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

Invariants

Base field:	$\F_{3}$
Dimension:	$4$
L-polynomial:	$1 - x + 3 x^{2} - 5 x^{3} + 16 x^{4} - 15 x^{5} + 27 x^{6} - 27 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.172732979144$, $\pm0.335169663346$, $\pm0.627267020856$, $\pm0.735169663346$
Angle rank:	$2$ (numerical)
Number field:	8.0.58140625.2
Galois group:	$C_2^2:C_4$
Jacobians:	$2$
Isomorphism classes:	102
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$80$	$14080$	$396080$	$68710400$	$3844000000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$15$	$21$	$119$	$268$	$675$	$2271$	$6719$	$19893$	$58850$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 1 is hyperelliptic):

• $y^2=2 x^{10}+2 x^9+x^5+x^2+2 x$
• $x y+t^2=y^3+x^2 z-y^2 z+y z^2+z^3+x y t+y^2 t=0$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{5}}$.

Endomorphism algebra over $\F_{3}$

The endomorphism algebra of this simple isogeny class is 8.0.58140625.2.

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

The base change of $A$ to $\F_{3^{5}}$ is 2.243.m_w^2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.1525.1$)$

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.b_d_f_q	$2$	(not in LMFDB)
4.3.ag_x_aci_er	$5$	(not in LMFDB)
4.3.e_d_ak_abd	$5$	(not in LMFDB)