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

• Label: 2.163.abt_bfu
• 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:	$( 1 - 25 x + 163 x^{2} )( 1 - 20 x + 163 x^{2} )$
	$1 - 45 x + 826 x^{2} - 7335 x^{3} + 26569 x^{4}$
Frobenius angles:	$\pm0.0652307277549$, $\pm0.213555132351$
Angle rank:	$2$ (numerical)
Jacobians:	$40$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$20016$	$696076416$	$18748356375744$	$498319460689752576$	$13239671720157033432336$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$119$	$26197$	$4329128$	$705923161$	$115063927769$	$18755372172598$	$3057125227966523$	$498311413555096753$	$81224760521236925144$	$13239635966914544781397$

Jacobians and polarizations

This isogeny class contains the Jacobians of 40 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=39x^6+52x^5+144x^4+150x^3+65x^2+53x+75$
• $y^2=40x^6+60x^5+40x^4+80x^3+160x^2+11x+128$
• $y^2=154x^6+32x^5+85x^4+3x^3+63x^2+14x+78$
• $y^2=13x^6+58x^5+160x^4+154x^3+56x^2+95x+118$
• $y^2=28x^6+30x^5+132x^4+42x^3+130x^2+98x$
• $y^2=86x^6+114x^5+50x^4+57x^3+79x^2+57x+50$
• $y^2=47x^6+107x^5+132x^4+71x^3+145x^2+75x+40$
• $y^2=33x^6+131x^5+128x^4+160x^3+139x^2+141x+116$
• $y^2=7x^6+78x^5+15x^4+39x^3+125x^2+30x+136$
• $y^2=50x^6+150x^5+106x^4+130x^3+105x^2+138x+157$
• $y^2=138x^6+123x^5+138x^4+42x^3+63x^2+161x+58$
• $y^2=7x^6+2x^5+118x^4+2x^3+11x^2+31x+140$
• $y^2=132x^6+161x^5+39x^4+78x^3+17x^2+95x+20$
• $y^2=70x^6+66x^5+39x^4+68x^3+137x^2+96x+58$
• $y^2=20x^6+161x^5+28x^4+162x^3+133x^2+60x+41$
• $y^2=51x^6+110x^5+42x^4+66x^3+134x^2+56x+24$
• $y^2=52x^6+45x^5+23x^4+31x^3+108x^2+91x+106$
• $y^2=157x^6+12x^5+17x^4+103x^3+128x^2+36x+143$
• $y^2=74x^6+6x^5+89x^4+93x^3+4x^2+85x+102$
• $y^2=114x^6+51x^5+40x^4+157x^3+18x^2+98x+45$
• and

• $y^2=37x^6+77x^5+43x^4+42x^3+82x^2+154x+6$
• $y^2=120x^6+108x^5+111x^4+144x^3+87x+29$
• $y^2=7x^6+110x^5+13x^4+81x^3+86x^2+112x+10$
• $y^2=72x^6+60x^5+53x^4+6x^3+62x^2+113x+99$
• $y^2=81x^6+161x^5+38x^4+22x^3+140x^2+160x+152$
• $y^2=135x^6+76x^5+159x^4+x^3+43x^2+128x+115$
• $y^2=25x^6+129x^5+55x^4+22x^3+84x^2+117x+29$
• $y^2=134x^6+65x^5+148x^4+132x^3+138x^2+67x+94$
• $y^2=72x^6+119x^5+120x^4+44x^3+71x^2+15x+114$
• $y^2=147x^6+90x^5+125x^4+143x^3+42x^2+57x+24$
• $y^2=158x^6+44x^5+78x^4+117x^3+39x^2+43x+32$
• $y^2=109x^6+117x^5+105x^4+111x^3+21x^2+84x+135$
• $y^2=102x^6+59x^5+54x^4+76x^3+54x^2+84x+126$
• $y^2=13x^6+159x^5+23x^4+90x^3+64x^2+37x+116$
• $y^2=32x^6+153x^5+5x^4+162x^3+136x^2+107x+16$
• $y^2=44x^6+21x^5+5x^4+131x^3+112x^2+75x+98$
• $y^2=33x^6+162x^4+139x^3+68x^2+9x+155$
• $y^2=151x^6+132x^5+58x^4+87x^3+53x^2+20x+9$
• $y^2=6x^6+100x^5+99x^4+136x^3+88x^2+151x$
• $y^2=12x^6+49x^5+113x^4+125x^3+90x^2+114x+121$

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.au 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.au : \(\Q(\sqrt{-7}) \).

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.af_ags	$2$	(not in LMFDB)
2.163.f_ags	$2$	(not in LMFDB)
2.163.bt_bfu	$2$	(not in LMFDB)
2.163.am_gk	$3$	(not in LMFDB)
2.163.ad_ao	$3$	(not in LMFDB)