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

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

Invariants

Base field:	$\F_{17}$
Dimension:	$2$
L-polynomial:	$( 1 - 8 x + 17 x^{2} )( 1 + 6 x + 17 x^{2} )$
	$1 - 2 x - 14 x^{2} - 34 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.0779791303774$, $\pm0.759367463010$
Angle rank:	$2$ (numerical)
Jacobians:	$16$
Isomorphism classes:	154

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$240$	$74880$	$23203440$	$6996787200$	$2012133433200$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$258$	$4720$	$83774$	$1417136$	$24138306$	$410366672$	$6975658366$	$118588882000$	$2015995263618$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.ai $\times$ 1.17.g and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.17.ai : \(\Q(\sqrt{-1}) \).
• 1.17.g : \(\Q(\sqrt{-2}) \).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.17.ao_de	$2$	(not in LMFDB)
2.17.c_ao	$2$	(not in LMFDB)
2.17.o_de	$2$	(not in LMFDB)