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

• Label: 2.17.d_g
• 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:	$( 1 - 4 x + 17 x^{2} )( 1 + 7 x + 17 x^{2} )$
	$1 + 3 x + 6 x^{2} + 51 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.338793663197$, $\pm0.822719357511$
Angle rank:	$2$ (numerical)
Jacobians:	$10$
Isomorphism classes:	122
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$350$	$84700$	$24764600$	$7026712000$	$2010892556750$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$21$	$293$	$5040$	$84129$	$1416261$	$24137426$	$410293317$	$6975902401$	$118588760640$	$2015992733093$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.ae $\times$ 1.17.h 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.ae : \(\Q(\sqrt{-13}) \).
• 1.17.h : \(\Q(\sqrt{-19}) \).

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.al_ck	$2$	(not in LMFDB)
2.17.ad_g	$2$	(not in LMFDB)
2.17.l_ck	$2$	(not in LMFDB)