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

• Label: 2.17.ah_bi
• Base field: $\F_{17}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• 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:	$( 1 - 7 x + 17 x^{2} )( 1 + 17 x^{2} )$
	$1 - 7 x + 34 x^{2} - 119 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.177280642489$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$12$
Cyclic group of points:	yes

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$198$	$89100$	$24216192$	$6956928000$	$2019167283078$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$11$	$309$	$4928$	$83297$	$1422091$	$24157026$	$410368123$	$6975632833$	$118587672896$	$2015994593589$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.ah $\times$ 1.17.a 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.ah : \(\Q(\sqrt{-19}) \).
• 1.17.a : \(\Q(\sqrt{-17}) \).

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

The base change of $A$ to $\F_{17^{2}}$ is 1.289.ap $\times$ 1.289.bi. The endomorphism algebra for each factor is:

• 1.289.ap : \(\Q(\sqrt{-19}) \).
• 1.289.bi : the quaternion algebra over \(\Q\) ramified at $17$ and $\infty$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.17.h_bi	$2$	(not in LMFDB)