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

• Label: 2.163.abs_bfe
• 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:	$( 1 - 22 x + 163 x^{2} )^{2}$
	$1 - 44 x + 810 x^{2} - 7172 x^{3} + 26569 x^{4}$
Frobenius angles:	$\pm0.169471200781$, $\pm0.169471200781$
Angle rank:	$1$ (numerical)
Jacobians:	$32$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$20164$	$697593744$	$18756331016164$	$498351193239684096$	$13239774480732511599364$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$120$	$26254$	$4330968$	$705968110$	$115064820840$	$18755386876798$	$3057125425570056$	$498311415554219614$	$81224760530998111704$	$13239635966753852127214$

Jacobians and polarizations

This isogeny class contains the Jacobians of 32 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=116x^6+154x^5+28x^4+57x^3+28x^2+154x+116$
• $y^2=36x^6+135x^5+117x^4+122x^3+16x^2+20x+114$
• $y^2=129x^6+15x^4+15x^2+129$
• $y^2=63x^6+78x^5+28x^4+160x^3+2x^2+32x+52$
• $y^2=70x^6+49x^5+95x^4+144x^3+7x^2+67x+127$
• $y^2=115x^6+79x^5+89x^4+77x^3+89x^2+79x+115$
• $y^2=68x^6+71x^5+157x^4+82x^3+23x^2+116x+39$
• $y^2=47x^6+57x^5+145x^4+67x^3+55x^2+96x+46$
• $y^2=80x^6+97x^4+97x^2+80$
• $y^2=156x^6+71x^5+41x^4+7x^3+41x^2+71x+156$
• $y^2=97x^6+132x^5+13x^4+34x^3+13x^2+132x+97$
• $y^2=33x^6+25x^5+9x^4+103x^3+9x^2+25x+33$
• $y^2=127x^6+38x^5+50x^4+43x^3+45x^2+121x+10$
• $y^2=117x^6+112x^5+48x^4+81x^3+106x^2+58x+98$
• $y^2=114x^6+130x^5+94x^4+30x^3+130x^2+54x+52$
• $y^2=55x^6+155x^5+105x^4+36x^3+105x^2+155x+55$
• $y^2=71x^6+74x^5+61x^4+89x^3+150x^2+72x+32$
• $y^2=75x^6+29x^5+32x^4+x^3+162x^2+126x+24$
• $y^2=2x^6+66x^3+154$
• $y^2=90x^6+45x^5+13x^4+16x^3+92x^2+162x+119$
• and

• $y^2=116x^6+113x^5+2x^4+139x^3+101x^2+19x+79$
• $y^2=142x^6+28x^5+123x^4+81x^3+139x^2+60x+75$
• $y^2=31x^6+99x^5+74x^4+70x^3+130x^2+137x+20$
• $y^2=9x^6+157x^5+119x^4+16x^3+147x^2+21x+30$
• $y^2=39x^6+72x^5+20x^4+150x^3+37x^2+156x+155$
• $y^2=2x^6+20x^3+44$
• $y^2=45x^6+22x^5+129x^4+135x^3+108x^2+90x+130$
• $y^2=156x^6+113x^5+49x^4+12x^3+140x^2+x+33$
• $y^2=82x^6+49x^5+92x^4+78x^3+3x^2+52x+79$
• $y^2=47x^6+37x^5+69x^4+65x^3+29x^2+53x+1$
• $y^2=107x^6+112x^5+11x^4+63x^3+51x^2+39x+130$
• $y^2=114x^6+119x^5+127x^4+147x^3+x^2+148x+8$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{163}$

The isogeny class factors as 1.163.aw^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-42}) \)$)$

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.a_agc	$2$	(not in LMFDB)
2.163.bs_bfe	$2$	(not in LMFDB)
2.163.w_mj	$3$	(not in LMFDB)