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

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

Invariants

Base field:	$\F_{17}$
Dimension:	$2$
L-polynomial:	$1 - 16 x^{2} + 289 x^{4}$
Frobenius angles:	$\pm0.172020869623$, $\pm0.827979130377$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\zeta_{8})\)
Galois group:	$C_2^2$
Jacobians:	$5$

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$274$	$75076$	$24147346$	$7029816336$	$2015992088914$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$18$	$258$	$4914$	$84166$	$1419858$	$24157122$	$410338674$	$6975884158$	$118587876498$	$2015990277378$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{17^{2}}$.

Endomorphism algebra over $\F_{17}$

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{8})\).

Endomorphism algebra over $\overline{\F}_{17}$

The base change of $A$ to $\F_{17^{2}}$ is 1.289.aq^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.a_q	$4$	(not in LMFDB)
2.17.aq_du	$8$	(not in LMFDB)
2.17.ak_by	$8$	(not in LMFDB)