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

• Label: 2.17.a_p
• 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:	$1 + 15 x^{2} + 289 x^{4}$
Frobenius angles:	$\pm0.322719357511$, $\pm0.677280642489$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{19})\)
Galois group:	$C_2^2$
Jacobians:	$21$
Isomorphism classes:	13
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$305$	$93025$	$24127940$	$7035015625$	$2015996047025$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$18$	$320$	$4914$	$84228$	$1419858$	$24118310$	$410338674$	$6975842308$	$118587876498$	$2015998193600$

Jacobians and polarizations

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

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

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 \(\Q(i, \sqrt{19})\).

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

The base change of $A$ to $\F_{17^{2}}$ is 1.289.p^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.ao_df	$4$	(not in LMFDB)
2.17.a_ap	$4$	(not in LMFDB)
2.17.o_df	$4$	(not in LMFDB)