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

• Label: 2.17.e_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:	$1 + 4 x + 32 x^{2} + 68 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.482640758168$, $\pm0.681416914314$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-58 +4 \sqrt{6}})\)
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	12
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:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$394$	$98500$	$23574202$	$6960010000$	$2016058773754$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$338$	$4798$	$83334$	$1419902$	$24137426$	$410384150$	$6975631486$	$118586922166$	$2015998697618$

Jacobians and polarizations

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

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

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 \(\Q(\sqrt{-58 +4 \sqrt{6}})\).

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.ae_bg	$2$	(not in LMFDB)