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

• Label: 2.17.g_bq
• 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:	$( 1 + 2 x + 17 x^{2} )( 1 + 4 x + 17 x^{2} )$
	$1 + 6 x + 42 x^{2} + 102 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.577979130377$, $\pm0.661206336803$
Angle rank:	$2$ (numerical)
Jacobians:	$8$
Isomorphism classes:	30
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$440$	$98560$	$23010680$	$6970163200$	$2021119806200$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$338$	$4680$	$83454$	$1423464$	$24128786$	$410314488$	$6975923326$	$118587750840$	$2015992692818$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.c $\times$ 1.17.e 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.c : \(\Q(\sqrt{-1}) \).
• 1.17.e : \(\Q(\sqrt{-13}) \).

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.ag_bq	$2$	(not in LMFDB)
2.17.ac_ba	$2$	(not in LMFDB)
2.17.c_ba	$2$	(not in LMFDB)