Abelian variety isogeny class 3.13.ae_bh_adi over $\F_{13}$

• Label: 3.13.ae_bh_adi
• Base field: $\F_{13}$
• 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:	$\F_{13}$
Dimension:	$3$
L-polynomial:	$1 - 4 x + 33 x^{2} - 86 x^{3} + 429 x^{4} - 676 x^{5} + 2197 x^{6}$
Frobenius angles:	$\pm0.288974508800$, $\pm0.419388285862$, $\pm0.600764434597$
Angle rank:	$3$ (numerical)
Number field:	6.0.23517473472.1
Galois group:	$S_4\times C_2$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1894$	$6488844$	$10963057246$	$23313352321584$	$51222455194072534$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$220$	$2272$	$28580$	$371560$	$4822672$	$62735578$	$815770844$	$10604466490$	$137857838920$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

• $y^2=2 x^8+6 x^7+12 x^6+8 x^5+11 x^4+10 x^3+3 x^2+3 x+4$
• $y^2=2 x^8+8 x^7+9 x^5+6 x^4+4 x^3+11 x^2+9 x+6$
• $y^2=2 x^8+8 x^7+2 x^6+9 x^5+11 x^4+6 x^3+12 x^2+5 x+8$
• $y^2=2 x^8+5 x^7+6 x^6+3 x^5+2 x^4+5 x^3+3 x+2$
• $y^2=2 x^8+7 x^7+x^6+7 x^5+x^4+10 x^2+8 x+11$
• $y^2=2 x^8+8 x^7+11 x^6+12 x^5+10 x^4+x^3+2 x^2+3 x+4$
• $y^2=x^8+12 x^7+4 x^6+5 x^5+6 x^4+6 x^3+3 x^2+5 x+7$
• $y^2=2 x^8+11 x^7+10 x^6+11 x^5+11 x^4+2 x^3+6 x^2+9 x+9$
• $y^2=2 x^8+2 x^7+x^6+7 x^5+3 x^4+7 x^3+7 x^2+5 x+6$
• $y^2=2 x^8+10 x^7+10 x^6+11 x^5+9 x^4+10 x^3+11 x^2+10 x+8$
• $y^2=2 x^8+7 x^7+7 x^6+4 x^5+6 x^4+x^3+6 x^2+3 x+2$
• $y^2=2 x^8+10 x^7+10 x^6+4 x^5+6 x^4+5 x^3+5 x^2+x+3$
• $y^2=x^8+x^7+10 x^6+9 x^5+2 x^4+6 x^3+2 x+2$
• $y^2=x^8+10 x^7+x^6+5 x^5+7 x^3+2 x^2+10 x+11$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13}$.

Endomorphism algebra over $\F_{13}$

The endomorphism algebra of this simple isogeny class is 6.0.23517473472.1.

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
3.13.e_bh_di	$2$	(not in LMFDB)