Abelian variety isogeny class 2.167.abu_bhf over $\F_{167}$

• Label: 2.167.abu_bhf
• Base field: $\F_{167}$
• 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:	$\F_{167}$
Dimension:	$2$
L-polynomial:	$( 1 - 23 x + 167 x^{2} )^{2}$
	$1 - 46 x + 863 x^{2} - 7682 x^{3} + 27889 x^{4}$
Frobenius angles:	$\pm0.150776270497$, $\pm0.150776270497$
Angle rank:	$1$ (numerical)
Jacobians:	$21$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21025$	$767013025$	$21685972512400$	$604994735268105625$	$16872061939432877175625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$122$	$27500$	$4656176$	$777831828$	$129893017342$	$21691979396750$	$3622557823702786$	$604967119441982308$	$101029508549971808912$	$16871927924916425577500$

Jacobians and polarizations

This isogeny class contains the Jacobians of 21 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=46x^6+160x^5+163x^4+161x^3+163x^2+160x+46$
• $y^2=37x^6+11x^5+130x^4+147x^3+130x^2+11x+37$
• $y^2=103x^6+89x^5+112x^4+145x^3+91x^2+56x+101$
• $y^2=4x^6+156x^5+26x^4+103x^3+26x^2+156x+4$
• $y^2=142x^6+63x^5+42x^4+128x^3+72x^2+166x+35$
• $y^2=50x^6+11x^5+83x^4+149x^3+136x^2+2x+26$
• $y^2=13x^6+31x^5+84x^4+34x^3+84x^2+31x+13$
• $y^2=37x^6+82x^5+125x^4+120x^3+125x^2+82x+37$
• $y^2=35x^6+37x^5+126x^4+92x^3+126x^2+37x+35$
• $y^2=21x^6+55x^5+139x^4+79x^3+139x^2+55x+21$
• $y^2=131x^6+152x^5+63x^4+41x^3+63x^2+152x+131$
• $y^2=44x^6+125x^5+130x^4+120x^3+146x^2+69x+156$
• $y^2=146x^6+53x^5+21x^4+72x^3+21x^2+53x+146$
• $y^2=13x^6+14x^5+71x^4+x^3+59x^2+151x+34$
• $y^2=86x^6+14x^5+16x^4+47x^3+16x^2+14x+86$
• $y^2=143x^6+64x^5+142x^4+98x^3+142x^2+64x+143$
• $y^2=84x^6+98x^5+24x^4+30x^3+123x^2+99x+91$
• $y^2=135x^6+33x^5+6x^4+114x^3+6x^2+33x+135$
• $y^2=68x^6+61x^5+16x^4+124x^3+53x^2+166x+107$
• $y^2=51x^6+60x^5+25x^4+25x^3+78x^2+152x+98$
• $y^2=20x^6+159x^5+104x^4+38x^3+81x^2+18x+139$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{167}$.

Endomorphism algebra over $\F_{167}$

The isogeny class factors as 1.167.ax^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-139}) \)$)$

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
2.167.a_ahn	$2$	(not in LMFDB)
2.167.bu_bhf	$2$	(not in LMFDB)
2.167.x_ny	$3$	(not in LMFDB)