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

• Label: 2.17.o_de
• 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:	$( 1 + 6 x + 17 x^{2} )( 1 + 8 x + 17 x^{2} )$
	$1 + 14 x + 82 x^{2} + 238 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.759367463010$, $\pm0.922020869623$
Angle rank:	$2$ (numerical)
Jacobians:	$2$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$624$	$74880$	$24206832$	$6996787200$	$2014424830704$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$32$	$258$	$4928$	$83774$	$1418752$	$24138306$	$410355040$	$6975658366$	$118588066016$	$2015995263618$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.g $\times$ 1.17.i 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.g : \(\Q(\sqrt{-2}) \).
• 1.17.i : \(\Q(\sqrt{-1}) \).

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.ao_de	$2$	(not in LMFDB)
2.17.ac_ao	$2$	(not in LMFDB)
2.17.c_ao	$2$	(not in LMFDB)