Abelian variety isogeny class 3.11.ac_n_au over $\F_{11}$

• Label: 3.11.ac_n_au
• Base field: $\F_{11}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{11}$
Dimension:	$3$
L-polynomial:	$1 - 2 x + 13 x^{2} - 20 x^{3} + 143 x^{4} - 242 x^{5} + 1331 x^{6}$
Frobenius angles:	$\pm0.226156762349$, $\pm0.444827652106$, $\pm0.716532152134$
Angle rank:	$3$ (numerical)
Number field:	6.0.1689725952.1
Galois group:	$S_4\times C_2$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1224$	$2144448$	$2376287064$	$3195604942848$	$4175122567275144$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$144$	$1342$	$14908$	$160970$	$1772496$	$19501982$	$214309820$	$2357738506$	$25937432784$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 77 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

• $y^2=2 x^8+7 x^7+x^6+4 x^5+7 x^4+6 x^2+5 x+5$
• $y^2=5 x^8+4 x^7+7 x^6+7 x^5+2 x^3+5 x^2+3 x+7$
• $y^2=4 x^8+9 x^7+5 x^6+9 x^5+5 x^4+5 x^3+7 x^2+3 x+5$
• $y^2=10 x^8+2 x^7+8 x^5+4 x^4+2 x^3+6 x^2+10 x$
• $y^2=7 x^8+2 x^7+3 x^6+7 x^5+x^4+3 x^3+9 x^2+10 x+4$
• $y^2=5 x^8+4 x^7+3 x^5+9 x^4+2 x^3+4 x^2+7 x$
• $y^2=6 x^8+6 x^7+4 x^6+4 x^5+2 x^4+5 x^3+7 x^2+6 x+10$
• $y^2=7 x^8+2 x^7+5 x^6+3 x^5+4 x^3+9$
• $y^2=8 x^8+2 x^7+4 x^6+10 x^5+7 x^4+10 x^3+8 x^2+x+2$
• $y^2=5 x^8+3 x^7+5 x^6+x^4+5 x^3+7 x^2+3 x+10$
• $y^2=2 x^8+3 x^7+7 x^5+8 x^4+4 x^3+7 x^2+8 x+4$
• $y^2=5 x^8+2 x^7+6 x^6+4 x^5+10 x^4+9 x^3+2 x^2+10 x+10$
• $y^2=10 x^8+x^7+6 x^6+3 x^5+6 x^4+10 x^3+6 x^2+x+7$
• $y^2=x^8+3 x^7+x^6+3 x^5+9 x^3+4 x+8$
• $y^2=4 x^8+9 x^7+6 x^6+x^5+7 x^4+9 x^3+3 x^2+7 x+8$
• $y^2=6 x^8+9 x^7+4 x^6+6 x^5+6 x^4+6 x^3+8 x^2+10 x+4$
• $y^2=6 x^8+6 x^7+5 x^6+8 x^5+6 x^4+6 x^3+4 x^2+3 x+10$
• $y^2=6 x^8+8 x^7+9 x^6+3 x^5+4 x^4+4 x^3+x^2+9 x+1$
• $y^2=8 x^8+10 x^7+2 x^6+6 x^5+6 x^4+8 x^2+8 x+6$
• $y^2=4 x^8+8 x^7+4 x^6+9 x^5+5 x^4+2 x^3+4 x+6$
• and

• $y^2=6 x^8+5 x^7+5 x^5+3 x^4+10 x^3+8 x^2+x+2$
• $y^2=10 x^8+3 x^7+7 x^6+4 x^5+6 x^4+4 x^3+7 x^2+8 x+3$
• $y^2=3 x^8+6 x^7+10 x^6+6 x^5+2 x^3+2 x^2+5 x+7$
• $y^2=7 x^8+5 x^7+4 x^6+10 x^5+4 x^4+5 x^3+4 x^2+6 x+2$
• $y^2=6 x^8+2 x^7+4 x^6+8 x^5+2 x^4+7 x^3+7 x+6$
• $y^2=x^8+3 x^7+7 x^6+5 x^5+5 x^4+6 x^3+4 x+9$
• $y^2=5 x^7+10 x^6+7 x^5+6 x^4+6 x^3+6 x^2+9 x+2$
• $y^2=x^8+x^7+6 x^6+8 x^5+8 x^4+x^3+6 x^2+2 x+10$
• $y^2=6 x^8+4 x^7+5 x^6+3 x^5+3 x^4+7 x^3+7 x^2+5 x+7$
• $y^2=5 x^8+9 x^6+3 x^5+9 x^4+2 x^3+7 x^2+6 x+9$
• $y^2=5 x^8+6 x^7+5 x^6+4 x^5+5 x^4+6 x^3+x^2+4 x+6$
• $y^2=2 x^8+x^7+3 x^6+6 x^5+9 x^3+4 x^2+7 x+1$
• $y^2=3 x^8+3 x^7+2 x^5+x^4+6 x^3+2 x+10$
• $y^2=x^8+2 x^7+6 x^6+3 x^5+10 x^4+7 x^3+x^2+6 x+8$
• $y^2=3 x^8+9 x^7+6 x^6+10 x^4+2 x^2+3 x+10$
• $y^2=3 x^8+2 x^7+3 x^6+2 x^5+6 x^4+8 x^3+x+7$
• $y^2=4 x^7+6 x^6+7 x^5+8 x^3+6 x^2+x+3$
• $y^2=2 x^8+10 x^7+2 x^6+9 x^5+x^4+3 x^3+10 x^2+2 x+9$
• $y^2=x^8+6 x^7+2 x^5+5 x^4+3 x^3+5 x^2+7$
• $y^2=4 x^8+7 x^7+3 x^6+x^5+7 x^4+6 x^3+2 x^2+9 x+2$
• $y^2=10 x^8+9 x^7+5 x^6+x^5+4 x^4+6 x^3+x^2+4 x$
• $y^2=10 x^8+8 x^7+2 x^6+x^3+9 x^2+9 x$
• $y^2=2 x^7+x^6+3 x^5+10 x^4+8 x^3+5 x^2+6$
• $y^2=10 x^8+10 x^7+5 x^6+2 x^5+5 x^4+10 x^3+2 x^2+7 x+2$
• $y^2=10 x^8+4 x^7+8 x^6+2 x^5+6 x^4+8 x^2+8 x+8$
• $y^2=6 x^8+6 x^7+9 x^6+3 x^5+3 x^3+x^2+7 x+4$
• $y^2=8 x^8+5 x^7+7 x^6+2 x^5+6 x^4+4 x^3+8 x^2+3 x+9$
• $y^2=4 x^8+9 x^7+5 x^6+2 x^5+2 x^4+10 x^3+x^2+10$
• $y^2=x^8+x^7+x^6+6 x^5+4 x^4+6 x^2+6 x+5$
• $y^2=9 x^8+5 x^7+8 x^6+4 x^4+6 x^3+8 x^2+10 x+2$
• $y^2=6 x^8+4 x^6+7 x^5+6 x^4+8 x^3+4 x^2+3 x+8$
• $y^2=5 x^8+2 x^7+2 x^6+7 x^5+9 x^4+4 x^3+2 x^2+2 x+6$
• $y^2=10 x^8+7 x^7+5 x^6+4 x^5+3 x^4+6 x^3+7 x^2+5 x+2$
• $y^2=4 x^8+x^7+x^6+2 x^5+10 x^4+6 x^3+6 x^2+9 x+7$
• $y^2=3 x^7+3 x^6+9 x^5+x^4+2 x^3+2 x^2+3 x+6$
• $y^2=10 x^8+9 x^7+7 x^6+x^5+2 x^4+8 x^3+10 x^2+5 x+9$
• $y^2=2 x^8+2 x^7+8 x^6+9 x^5+10 x^4+3 x^3+x^2+4 x+2$
• $y^2=4 x^8+4 x^7+9 x^6+10 x^5+3 x^4+9 x^3+4 x^2+x$
• $y^2=x^8+7 x^7+10 x^6+8 x^5+6 x^4+8 x^3+7 x^2+3 x+4$
• $y^2=10 x^8+2 x^7+5 x^6+10 x^5+4 x^4+6 x^3+6 x^2+6 x+5$
• $y^2=7 x^8+8 x^6+10 x^5+2 x^4+3 x^3+9 x^2+3 x+10$
• $y^2=x^8+6 x^7+7 x^6+2 x^5+x^3+6 x^2+6 x+10$
• $y^2=6 x^8+7 x^7+7 x^6+7 x^5+10 x^4+2 x^2+9 x+2$
• $y^2=4 x^8+x^7+7 x^6+5 x^5+9 x^4+x^2+4 x+8$
• $y^2=6 x^8+3 x^7+6 x^6+7 x^5+7 x^4+4 x^3+5 x^2+7 x+10$
• $y^2=10 x^8+x^7+7 x^6+4 x^5+6 x^4+4 x^3+8 x^2+x+1$
• $y^2=x^8+5 x^6+10 x^5+5 x^4+7 x^3+4 x^2+10 x+10$
• $y^2=5 x^7+6 x^6+5 x^5+7 x^4+4 x^3+10 x^2+2$
• $y^2=6 x^8+8 x^7+9 x^6+4 x^5+7 x^4+10 x^3+4 x+10$
• $y^2=6 x^8+10 x^7+9 x^6+10 x^5+2 x^4+x^3+4 x^2+2 x+10$
• $y^2=2 x^8+8 x^6+10 x^5+3 x^4+3 x^3+10 x^2+6 x+5$
• $y^2=5 x^8+8 x^7+8 x^6+4 x^5+4 x^4+8 x^3+x^2+3 x+6$
• $y^2=9 x^8+6 x^7+10 x^6+6 x^5+6 x^4+5 x^3+6 x^2+9 x+6$
• $y^2=8 x^8+4 x^7+6 x^6+x^5+2 x^2+x+4$
• $y^2=6 x^8+3 x^7+9 x^6+3 x^5+x^4+10 x^2+4 x+9$
• $y^2=8 x^8+3 x^7+4 x^6+x^4+7 x^3+7 x^2+10 x+10$
• $y^2=10 x^8+x^7+8 x^6+3 x^5+7 x^4+x^3+2 x^2+3 x+2$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11}$.

Endomorphism algebra over $\F_{11}$

The endomorphism algebra of this simple isogeny class is 6.0.1689725952.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
3.11.c_n_u	$2$	(not in LMFDB)