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

• Label: 4.3.ae_s_abo_dn
• 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:	$( 1 - x + 3 x^{2} )^{4}$
	$1 - 4 x + 18 x^{2} - 40 x^{3} + 91 x^{4} - 120 x^{5} + 162 x^{6} - 108 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.406785250661$, $\pm0.406785250661$, $\pm0.406785250661$, $\pm0.406785250661$
Angle rank:	$1$ (numerical)
Jacobians:	$1$
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$81$	$50625$	$1679616$	$31640625$	$2058346161$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$30$	$60$	$54$	$120$	$690$	$2520$	$7014$	$19140$	$57150$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.ab^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-11}) \)$)$

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
4.3.ac_m_aq_cb	$2$	(not in LMFDB)
4.3.a_k_a_br	$2$	(not in LMFDB)
4.3.c_m_q_cb	$2$	(not in LMFDB)
4.3.e_s_bo_dn	$2$	(not in LMFDB)
4.3.ab_d_i_ai	$3$	(not in LMFDB)
4.3.c_ad_c_bc	$3$	(not in LMFDB)