Abelian variety isogeny class 2.17.d_bg over $\F_{17}$

• Label: 2.17.d_bg
• Base field: $\F_{17}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{17}$
Dimension:	$2$
L-polynomial:	$1 + 3 x + 32 x^{2} + 51 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.478306855228$, $\pm0.642157270347$
Angle rank:	$2$ (numerical)
Number field:	4.0.270504.1
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	12
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:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$376$	$100768$	$23612800$	$6939690624$	$2017532949496$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$21$	$345$	$4806$	$83089$	$1420941$	$24137790$	$410356653$	$6975766369$	$118586950182$	$2015995413225$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

• $y^2=12 x^6+9 x^5+x^4+12 x^3+2 x^2+12 x+5$
• $y^2=11 x^6+x^5+x^4+2 x^3+3 x^2+14 x+1$
• $y^2=12 x^6+12 x^5+15 x^4+16 x^3+2 x^2+16 x$
• $y^2=6 x^6+10 x^5+6 x^3+11 x^2+5 x+1$
• $y^2=4 x^6+14 x^5+9 x^4+6 x^3+9 x^2+4 x+3$
• $y^2=13 x^6+14 x^5+8 x^4+16 x^3+4 x^2+15 x+4$
• $y^2=4 x^6+7 x^5+x^4+5 x^3+12 x^2+8 x$
• $y^2=2 x^6+3 x^5+15 x^4+6 x^3+7 x^2+9 x+9$
• $y^2=5 x^6+x^5+4 x^4+12 x^3+9 x^2+10 x+10$
• $y^2=13 x^6+15 x^5+16 x^4+12 x^3+3 x^2+10 x+9$
• $y^2=14 x^6+9 x^5+4 x^4+14 x^3+5 x^2+4 x+6$
• $y^2=16 x^6+10 x^4+15 x^3+4 x^2+13 x+12$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{17}$.

Endomorphism algebra over $\F_{17}$

The endomorphism algebra of this simple isogeny class is 4.0.270504.1.

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
2.17.ad_bg	$2$	(not in LMFDB)