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

• Label: 2.17.ab_bc
• 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:	$( 1 - 3 x + 17 x^{2} )( 1 + 2 x + 17 x^{2} )$
	$1 - x + 28 x^{2} - 17 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.381477984739$, $\pm0.577979130377$
Angle rank:	$2$ (numerical)
Jacobians:	$21$
Isomorphism classes:	138

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$300$	$100800$	$24292800$	$6945120000$	$2015933407500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$17$	$345$	$4946$	$83153$	$1419817$	$24132510$	$410319241$	$6975985633$	$118588574642$	$2015989341225$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 21 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.ad $\times$ 1.17.c 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.ad : \(\Q(\sqrt{-59}) \).
• 1.17.c : \(\Q(\sqrt{-1}) \).

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.af_bo	$2$	(not in LMFDB)
2.17.b_bc	$2$	(not in LMFDB)
2.17.f_bo	$2$	(not in LMFDB)