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

• Label: 2.163.abt_bfw
• 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:	$1 - 45 x + 828 x^{2} - 7335 x^{3} + 26569 x^{4}$
Frobenius angles:	$\pm0.0881433563308$, $\pm0.204603807563$
Angle rank:	$2$ (numerical)
Number field:	4.0.3299224.1
Galois group:	$D_{4}$
Jacobians:	$14$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$20018$	$696186004$	$18749527641416$	$498326226814682656$	$13239699370092866909918$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$119$	$26201$	$4329398$	$705932745$	$115064168069$	$18755376901922$	$3057125304321263$	$498311414577953809$	$81224760532375326674$	$13239635967004781865161$

Jacobians and polarizations

This isogeny class contains the Jacobians of 14 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=128x^6+91x^5+157x^4+128x^3+56x^2+53x+140$
• $y^2=8x^6+77x^5+19x^4+138x^3+38x^2+18x+78$
• $y^2=27x^6+92x^5+121x^4+108x^3+48x^2+23x+133$
• $y^2=162x^6+14x^5+19x^4+40x^3+82x^2+130x+107$
• $y^2=126x^6+61x^5+63x^4+60x^3+29x^2+66x+44$
• $y^2=6x^6+21x^5+7x^4+39x^3+127x^2+98x+68$
• $y^2=138x^6+127x^5+95x^4+26x^3+95x^2+136x+79$
• $y^2=146x^6+30x^5+34x^4+78x^3+x^2+10x+63$
• $y^2=12x^6+40x^5+25x^4+37x^3+139x^2+90x+42$
• $y^2=82x^6+100x^5+132x^4+145x^3+107x^2+48x+159$
• $y^2=13x^6+5x^5+90x^4+135x^3+2x^2+136x+111$
• $y^2=10x^6+87x^5+121x^4+120x^3+57x^2+62x+30$
• $y^2=13x^6+28x^5+96x^4+106x^3+65x^2+90x+114$
• $y^2=69x^6+81x^5+160x^4+96x^3+10x^2+42x+108$

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 4.0.3299224.1.

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.bt_bfw	$2$	(not in LMFDB)