Abelian variety isogeny class 2.211.aca_bqg over $\F_{211}$

• Label: 2.211.aca_bqg
• Base field: $\F_{211}$
• 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:	$\F_{211}$
Dimension:	$2$
L-polynomial:	$( 1 - 26 x + 211 x^{2} )^{2}$
	$1 - 52 x + 1098 x^{2} - 10972 x^{3} + 44521 x^{4}$
Frobenius angles:	$\pm0.147206137144$, $\pm0.147206137144$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$34596$	$1959655824$	$88224954838596$	$3928894709878785024$	$174914723241827175291876$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$160$	$44014$	$9391696$	$1982168494$	$418228949200$	$88245974708638$	$18619893805837120$	$3928797485115580894$	$828976268000541832576$	$174913992535554624347854$

Jacobians and polarizations

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

• $y^2=98x^6+101x^5+161x^4+15x^3+12x^2+41x+100$
• $y^2=192x^6+161x^5+147x^4+120x^3+15x^2+168x+57$
• $y^2=33x^6+42x^5+94x^4+155x^3+33x^2+119x+130$
• $y^2=169x^6+53x^5+170x^4+140x^3+183x^2+90x+106$
• $y^2=131x^6+197x^4+197x^2+131$
• $y^2=87x^6+167x^5+131x^4+126x^3+131x^2+167x+87$
• $y^2=110x^6+157x^5+107x^4+89x^3+166x^2+13x+41$
• $y^2=105x^6+124x^4+124x^2+105$
• $y^2=78x^6+84x^5+21x^4+205x^3+203x^2+137x+58$
• $y^2=171x^6+203x^5+67x^4+58x^3+67x^2+203x+171$
• $y^2=32x^6+80x^5+46x^4+173x^3+46x^2+80x+32$
• $y^2=x^6+x^3+87$
• $y^2=124x^6+129x^5+139x^4+136x^3+184x^2+61x+206$
• $y^2=128x^6+11x^5+34x^4+139x^3+34x^2+11x+128$
• $y^2=146x^6+161x^5+204x^4+208x^3+30x^2+136x+39$
• $y^2=166x^6+71x^5+159x^4+58x^3+93x^2+159x+172$
• $y^2=208x^6+207x^5+199x^4+196x^3+199x^2+207x+208$
• $y^2=89x^6+126x^5+131x^4+181x^3+26x^2+197x+90$
• $y^2=42x^6+190x^5+145x^4+107x^3+106x^2+133x+18$
• $y^2=26x^6+35x^5+83x^4+96x^3+103x^2+89x+75$
• and

• $y^2=30x^6+136x^5+145x^4+206x^3+15x^2+69x+128$
• $y^2=50x^6+158x^5+28x^4+121x^3+21x^2+29x+47$
• $y^2=180x^6+170x^5+79x^4+132x^3+178x^2+37x+16$
• $y^2=84x^6+190x^5+63x^4+52x^3+198x^2+174x+116$
• $y^2=21x^6+208x^5+31x^4+81x^3+30x^2+5x+23$
• $y^2=57x^6+164x^5+189x^4+47x^3+189x^2+164x+57$
• $y^2=85x^6+77x^5+188x^4+175x^3+157x^2+2x+51$
• $y^2=33x^6+110x^5+7x^4+199x^3+185x^2+140x+71$
• $y^2=37x^6+204x^5+40x^4+154x^3+59x^2+100x+168$
• $y^2=19x^6+198x^5+76x^4+133x^3+5x^2+20x+160$
• $y^2=x^6+100x^3+71$
• $y^2=138x^6+195x^5+11x^4+90x^3+93x^2+34x+31$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{211}$.

Endomorphism algebra over $\F_{211}$

The isogeny class factors as 1.211.aba^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
2.211.a_aju	$2$	(not in LMFDB)
2.211.ca_bqg	$2$	(not in LMFDB)
2.211.ba_rx	$3$	(not in LMFDB)