Abelian Variety Isogeny Class 2.167.a_hi over $\F_{167}$

• Label: 2.167.a_hi
• Base Field: $\F_{167}$
• Dimension: $2$
• Ordinary: No
• $p$-rank: $2$
• Principally polarizable: Yes
• Contains a Jacobian: Yes

Invariants

Base field:	$\F_{167}$
Dimension:	$2$
L-polynomial:	$( 1 - 12 x + 167 x^{2} )( 1 + 12 x + 167 x^{2} )$
Frobenius angles:	$\pm0.346308130027$, $\pm0.653691869973$
Angle rank:	$1$ (numerical)
Jacobians:	6375

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobians of 6375 curves, and hence is principally polarizable:

• $y^2=75x^6+147x^5+47x^4+128x^3+157x^2+115x+117$
• $y^2=41x^6+67x^5+68x^4+139x^3+117x^2+74x+84$
• $y^2=118x^6+65x^5+151x^4+75x^3+59x^2+119x+40$
• $y^2=131x^6+156x^5+39x^4+159x^3+111x^2+158x+149$
• $y^2=154x^6+112x^5+28x^4+127x^3+54x^2+122x+77$
• $y^2=79x^6+19x^5+47x^4+132x^3+139x^2+90x+142$
• $y^2=4x^6+47x^5+46x^4+92x^3+132x^2+96x+123$
• $y^2=20x^6+68x^5+63x^4+126x^3+159x^2+146x+114$
• $y^2=10x^6+37x^5+8x^4+95x^3+137x^2+12x+42$
• $y^2=50x^6+18x^5+40x^4+141x^3+17x^2+60x+43$
• $y^2=140x^6+103x^5+104x^4+77x^3+164x^2+154x+87$
• $y^2=32x^6+14x^5+19x^4+51x^3+152x^2+102x+101$
• $y^2=17x^6+62x^5+48x^4+30x^3+117x^2+103x+111$
• $y^2=57x^6+80x^5+45x^4+24x^3+50x^2+65x+64$
• $y^2=118x^6+66x^5+58x^4+120x^3+83x^2+158x+153$
• $y^2=148x^6+130x^5+36x^4+60x^3+87x^2+101x+78$
• $y^2=72x^6+149x^5+13x^4+133x^3+101x^2+4x+56$
• $y^2=24x^6+56x^5+45x^4+73x^3+3x^2+134x+12$
• $y^2=120x^6+113x^5+58x^4+31x^3+15x^2+2x+60$
• $y^2=137x^6+9x^5+95x^4+102x^3+144x^2+44x+88$
• and 6355 more

Point counts of the abelian variety

$r$	1	2	3	4	5	6	7	8	9	10
$A(\F_{q^r})$	28080	788486400	21691952558640	604997729856000000	16871927924959158308400	470540805806288442638649600	13122923468187020838031023741360	365985214017917247315715286016000000	10206961594320420340222156756888886201520	284661951905016649470981112183772749510560000

Point counts of the curve

$r$	1	2	3	4	5	6	7	8	9	10
$C(\F_{q^r})$	168	28270	4657464	777835678	129891985608	21691943520910	3622557586593624	604967119297872958	101029508532509551848	16871927924989221458350

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{167}$

The isogeny class factors as 1.167.am $\times$ 1.167.m and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.167.am : \(\Q(\sqrt{-131}) \).
• 1.167.m : \(\Q(\sqrt{-131}) \).

Endomorphism algebra over $\overline{\F}_{167}$

The base change of $A$ to $\F_{167^{2}}$ is 1.27889.hi^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-131}) \)$)$

All geometric endomorphisms are defined over $\F_{167^{2}}$.

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.ay_sk	$2$	(not in LMFDB)
2.167.y_sk	$2$	(not in LMFDB)
2.167.a_ahi	$4$	(not in LMFDB)