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

• Label: 2.163.abv_bhs
• 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:	$( 1 - 25 x + 163 x^{2} )( 1 - 22 x + 163 x^{2} )$
	$1 - 47 x + 876 x^{2} - 7661 x^{3} + 26569 x^{4}$
Frobenius angles:	$\pm0.0652307277549$, $\pm0.169471200781$
Angle rank:	$2$ (numerical)
Jacobians:	$6$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$19738$	$693869652$	$18741129704584$	$498305704588028064$	$13239664678055109160198$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$117$	$26113$	$4327458$	$705903673$	$115063866567$	$18755375328898$	$3057125318367237$	$498311415032988721$	$81224760537295546614$	$13239635966991965188993$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=140x^6+162x^5+12x^4+133x^3+4x^2+117x+36$
• $y^2=161x^6+10x^5+97x^4+13x^3+95x^2+136x+25$
• $y^2=19x^6+61x^5+118x^4+146x^3+114x^2+61x+103$
• $y^2=22x^6+138x^5+158x^4+78x^3+129x^2+53x+82$
• $y^2=94x^6+78x^5+12x^4+5x^3+82x^2+97x+75$
• $y^2=70x^6+118x^5+28x^4+3x^3+69x^2+x+149$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{163}$

The isogeny class factors as 1.163.az $\times$ 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.az : \(\Q(\sqrt{-3}) \).
• 1.163.aw : \(\Q(\sqrt{-42}) \).

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
2.163.ad_aiq	$2$	(not in LMFDB)
2.163.d_aiq	$2$	(not in LMFDB)
2.163.bv_bhs	$2$	(not in LMFDB)
2.163.ao_fu	$3$	(not in LMFDB)
2.163.af_abw	$3$	(not in LMFDB)