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

• Label: 2.17.ao_df
• 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:	$( 1 - 7 x + 17 x^{2} )^{2}$
	$1 - 14 x + 83 x^{2} - 238 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.177280642489$, $\pm0.177280642489$
Angle rank:	$1$ (numerical)
Jacobians:	$1$
Cyclic group of points:	no
Non-cyclic primes:	$11$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$121$	$75625$	$24285184$	$7035015625$	$2022342812281$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$260$	$4942$	$84228$	$1424324$	$24156830$	$410397572$	$6975842308$	$118587469294$	$2015989607300$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):

• $y^2=6 x^6+16 x^4+4 x^3+2 x^2+3$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.ah^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$

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_ap	$2$	(not in LMFDB)
2.17.o_df	$2$	(not in LMFDB)
2.17.h_bg	$3$	(not in LMFDB)