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

• Label: 2.163.abw_biq
• Base field: $\F_{163}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{163}$
Dimension:	$2$
L-polynomial:	$1 - 48 x + 900 x^{2} - 7824 x^{3} + 26569 x^{4}$
Frobenius angles:	$\pm0.0308818759184$, $\pm0.154480826596$
Angle rank:	$2$ (numerical)
Number field:	4.0.55552.1
Galois group:	$D_{4}$
Jacobians:	$4$

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$19598$	$692632516$	$18736047682094$	$498290274138211216$	$13239625854229675403198$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$116$	$26066$	$4326284$	$705881814$	$115063529156$	$18755370777026$	$3057125262158300$	$498311414352315358$	$81224760528362714708$	$13239635966855815968146$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=41x^6+85x^5+39x^4+160x^3+30x^2+101x+4$
• $y^2=123x^6+140x^5+63x^4+143x^3+59x^2+48x+19$
• $y^2=138x^6+140x^5+121x^4+52x^3+39x^2+159x+75$
• $y^2=92x^6+28x^5+122x^4+156x^3+7x^2+85x+111$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{163}$

The endomorphism algebra of this simple isogeny class is 4.0.55552.1.

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.bw_biq	$2$	(not in LMFDB)