Abelian variety isogeny class 2.163.abu_bgw over $\F_{163}$

• Label: 2.163.abu_bgw
• Base field: $\F_{163}$
• 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:	$\F_{163}$
Dimension:	$2$
L-polynomial:	$( 1 - 24 x + 163 x^{2} )( 1 - 22 x + 163 x^{2} )$
	$1 - 46 x + 854 x^{2} - 7498 x^{3} + 26569 x^{4}$
Frobenius angles:	$\pm0.110906256499$, $\pm0.169471200781$
Angle rank:	$2$ (numerical)
Jacobians:	$24$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$19880$	$695163840$	$18746811790280$	$498324695078246400$	$13239718531676564803400$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$118$	$26162$	$4328770$	$705930574$	$115064334598$	$18755382529154$	$3057125417782834$	$498311416260346846$	$81224760550450580950$	$13239635967102757019282$

Jacobians and polarizations

This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=108x^6+74x^5+77x^4+131x^3+6x^2+41x+70$
• $y^2=154x^6+56x^5+118x^4+12x^3+118x^2+56x+154$
• $y^2=139x^6+146x^5+134x^4+13x^3+140x^2+24x+52$
• $y^2=59x^6+43x^5+146x^4+2x^3+69x^2+53x+13$
• $y^2=159x^6+112x^5+30x^4+108x^3+129x^2+154x+66$
• $y^2=159x^6+84x^5+5x^4+44x^3+5x^2+84x+159$
• $y^2=157x^6+140x^5+155x^4+97x^3+155x^2+140x+157$
• $y^2=97x^6+16x^5+132x^4+133x^3+132x^2+16x+97$
• $y^2=92x^6+131x^5+7x^4+56x^3+7x^2+131x+92$
• $y^2=108x^6+70x^5+72x^4+46x^3+72x^2+70x+108$
• $y^2=94x^6+92x^5+26x^4+16x^3+26x^2+92x+94$
• $y^2=69x^6+126x^5+140x^4+118x^3+38x^2+115x+90$
• $y^2=55x^6+78x^5+29x^4+45x^3+29x^2+78x+55$
• $y^2=8x^6+29x^5+76x^4+158x^3+112x^2+128x+5$
• $y^2=37x^6+153x^5+36x^4+123x^3+36x^2+153x+37$
• $y^2=147x^6+92x^5+18x^4+77x^3+50x^2+78x+45$
• $y^2=28x^6+154x^5+44x^4+74x^3+73x^2+59x+5$
• $y^2=148x^6+135x^5+157x^4+63x^3+78x^2+158x+29$
• $y^2=66x^6+122x^5+38x^4+9x^3+38x^2+122x+66$
• $y^2=161x^6+161x^5+45x^4+32x^3+45x^2+161x+161$
• $y^2=105x^6+28x^5+122x^4+155x^3+106x^2+125x+110$
• $y^2=114x^6+150x^5+37x^4+x^3+37x^2+150x+114$
• $y^2=76x^6+81x^5+158x^4+81x^3+100x^2+126x+153$
• $y^2=28x^6+160x^5+129x^4+75x^3+129x^2+160x+28$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{163}$.

Endomorphism algebra over $\F_{163}$

The isogeny class factors as 1.163.ay $\times$ 1.163.aw and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.163.ay : \(\Q(\sqrt{-19}) \).
• 1.163.aw : \(\Q(\sqrt{-42}) \).

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
2.163.ac_ahu	$2$	(not in LMFDB)
2.163.c_ahu	$2$	(not in LMFDB)
2.163.bu_bgw	$2$	(not in LMFDB)