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

• Label: 3.13.a_ab_abg
• 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:	$1 - x^{2} - 32 x^{3} - 13 x^{4} + 2197 x^{6}$
Frobenius angles:	$\pm0.121524073204$, $\pm0.535983913158$, $\pm0.803371407109$
Angle rank:	$3$ (numerical)
Number field:	6.0.1285793024.1
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2152$	$4768832$	$10161823624$	$23254198255616$	$51163869166229032$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$168$	$2102$	$28508$	$371134$	$4836840$	$62745382$	$815739324$	$10605026030$	$137858649128$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 65 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

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

• $y^2=2 x^7+2 x^6+7 x^5+11 x^4+x^3+3 x^2+4 x+11$
• $y^2=2 x^8+3 x^7+5 x^6+7 x^5+7 x^4+9 x^3+9 x^2+5 x+5$
• $y^2=2 x^8+11 x^7+8 x^6+8 x^5+10 x^4+8 x^2+4 x+1$
• $y^2=x^8+6 x^6+10 x^5+10 x^4+6 x^3+12 x^2+11 x+10$
• $y^2=x^8+7 x^7+5 x^5+8 x^4+12 x^3+x^2+x+5$
• $y^2=x^8+11 x^7+3 x^6+x^5+6 x^4+2 x^3+8 x^2+11 x+9$
• $y^2=2 x^8+10 x^7+x^6+12 x^4+6 x^3+10 x^2+8$
• $y^2=2 x^8+4 x^7+12 x^6+3 x^4+11 x^3+3 x^2+8 x+6$
• $y^2=2 x^8+x^7+5 x^6+10 x^5+8 x^4+2 x^3+7 x^2+x$
• $y^2=x^7+8 x^6+7 x^5+3 x^4+9 x^3+11 x+12$
• $y^2=2 x^8+2 x^7+7 x^6+8 x^4+3 x^3+11 x^2+6$
• $y^2=x^8+5 x^7+9 x^6+7 x^5+2 x^4+10 x^3+9 x^2+4 x+3$
• $y^2=2 x^8+6 x^7+3 x^6+5 x^5+12 x^4+x^3+7 x^2+8 x+12$
• $y^2=x^8+8 x^7+4 x^6+2 x^5+6 x^4+9$
• $y^2=2 x^8+8 x^7+4 x^6+10 x^5+3 x^4+8 x^3+9 x^2+3 x+12$
• $y^2=x^8+x^7+12 x^6+8 x^5+10 x^4+9 x^2+3 x+1$
• $y^2=x^8+12 x^7+12 x^6+6 x^5+12 x^4+3 x^3+4 x^2+x+10$
• $y^2=x^8+2 x^6+12 x^5+3 x^4+3 x^3+9 x^2+2 x+5$
• $y^2=x^8+12 x^7+10 x^6+10 x^5+x^4+4 x^3+8 x^2+5 x+7$
• $y^2=x^8+7 x^7+4 x^6+2 x^5+11 x^4+5 x^3+4 x^2+6 x+5$
• $y^2=x^8+8 x^7+6 x^6+10 x^4+x^2+5 x+10$
• $y^2=x^8+9 x^7+2 x^6+8 x^5+11 x^4+x^3+12 x^2+4 x+1$
• $y^2=2 x^8+6 x^7+10 x^6+4 x^5+4 x^4+2 x^2+7 x+2$
• $y^2=2 x^8+3 x^7+11 x^6+5 x^5+2 x^4+12 x^3+12 x^2+6 x+9$
• $y^2=x^8+9 x^7+12 x^5+3 x^4+3 x^3+8 x^2+6 x+9$
• $y^2=2 x^8+10 x^7+10 x^5+8 x^4+6 x^3+8 x^2+x+8$
• $y^2=x^8+2 x^7+9 x^6+9 x^5+2 x^4+4 x^3+8 x^2+x+1$
• $y^2=2 x^8+8 x^7+2 x^6+2 x^5+4 x^3+x^2+10 x+4$
• $y^2=x^8+12 x^6+10 x^5+x^4+6 x^3+5 x^2+11 x+3$
• $y^2=2 x^8+7 x^7+10 x^6+4 x^5+5 x^4+4 x^3+x^2+3 x+2$
• $y^2=2 x^8+8 x^7+3 x^6+10 x^5+11 x^4+12 x^3+2 x^2+12 x+3$
• $y^2=x^8+8 x^7+9 x^6+10 x^5+6 x^4+11 x^3+12 x^2+10 x+2$
• $y^2=x^8+5 x^7+4 x^6+12 x^5+6 x^4+10 x^2+5 x+11$
• $y^2=2 x^8+6 x^7+x^6+4 x^5+6 x^4+4 x^3+6 x^2+7 x+8$
• $y^2=x^8+9 x^7+12 x^6+8 x^3+5 x^2+11 x+11$
• $y^2=x^8+2 x^7+9 x^5+11 x^4+8 x^3+x^2+11 x+7$
• $y^2=x^8+2 x^7+9 x^6+3 x^4+5 x^3+12 x^2+9 x+8$
• $y^2=2 x^8+9 x^7+4 x^6+11 x^5+7 x^4+10 x^3+2 x^2+10 x+9$
• $y^2=x^8+12 x^7+3 x^6+3 x^5+8 x^4+2 x^3+9 x^2+3$
• $y^2=2 x^8+4 x^7+4 x^6+4 x^5+8 x^4+x^3+2 x^2+7 x+3$
• $y^2=x^8+2 x^6+3 x^5+3 x^4+6 x^3+3 x^2+10 x+2$
• $y^2=2 x^8+10 x^7+4 x^6+10 x^5+2 x^4+6 x^3+11 x^2+12 x+12$
• $y^2=2 x^8+x^7+8 x^6+6 x^5+5 x^4+9 x^3+5 x^2+11 x+8$
• $y^2=x^8+10 x^7+11 x^6+x^5+8 x^4+7 x^3+8 x^2+5 x+2$
• $y^2=2 x^8+x^7+12 x^6+8 x^5+12 x^4+8 x^2+4 x+4$

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 6.0.1285793024.1.

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.a_ab_bg	$2$	(not in LMFDB)