Abelian variety isogeny class 2.139.abq_bbn over $\F_{139}$

• Label: 2.139.abq_bbn
• Base field: $\F_{139}$
• 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:	$\F_{139}$
Dimension:	$2$
L-polynomial:	$( 1 - 23 x + 139 x^{2} )( 1 - 19 x + 139 x^{2} )$
	$1 - 42 x + 715 x^{2} - 5838 x^{3} + 19321 x^{4}$
Frobenius angles:	$\pm0.0707251543800$, $\pm0.201746658314$
Angle rank:	$2$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$14157$	$366906969$	$7208491386096$	$139356427372718985$	$2692465343398114418277$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$98$	$18988$	$2684108$	$373308436$	$51889097918$	$7212552387766$	$1002544384691402$	$139353667103273956$	$19370159738716790612$	$2692452204156214591228$

Jacobians and polarizations

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

• $y^2=26x^6+124x^5+108x^4+96x^3+108x^2+124x+26$
• $y^2=43x^6+18x^5+74x^4+20x^3+3x^2+24x+109$
• $y^2=103x^6+44x^5+129x^4+71x^3+46x^2+42x+126$
• $y^2=22x^6+118x^5+49x^4+60x^3+49x^2+118x+22$
• $y^2=71x^6+90x^5+120x^4+17x^3+58x^2+16x+61$
• $y^2=127x^6+108x^5+59x^4+29x^3+130x^2+79x+123$
• $y^2=99x^6+27x^5+9x^4+127x^3+9x^2+27x+99$
• $y^2=114x^6+114x^5+77x^4+33x^3+77x^2+114x+114$
• $y^2=50x^6+59x^5+120x^4+32x^3+25x^2+8x+8$
• $y^2=17x^6+106x^5+45x^4+91x^3+45x^2+106x+17$
• $y^2=7x^6+60x^5+27x^4+41x^3+27x^2+60x+7$
• $y^2=62x^6+103x^5+6x^4+5x^3+116x^2+115x+71$
• $y^2=123x^6+66x^4+112x^3+60x^2+78x+37$
• $y^2=76x^6+97x^5+3x^4+66x^3+21x^2+95x+40$
• $y^2=86x^6+65x^5+26x^4+79x^3+9x^2+94x+16$
• $y^2=134x^6+42x^5+90x^4+8x^3+90x^2+42x+134$
• $y^2=35x^6+98x^5+5x^4+115x^3+102x^2+28x+94$
• $y^2=122x^6+102x^5+52x^4+122x^3+52x^2+102x+122$
• $y^2=84x^6+54x^5+108x^4+3x^3+108x^2+54x+84$
• $y^2=93x^6+61x^5+80x^4+111x^3+80x^2+61x+93$
• and

• $y^2=135x^6+45x^5+117x^4+116x^3+117x^2+45x+135$
• $y^2=128x^6+83x^5+5x^4+13x^3+104x^2+100x+91$
• $y^2=67x^6+84x^5+38x^4+120x^3+38x^2+84x+67$
• $y^2=104x^6+52x^5+75x^4+55x^3+45x^2+92x+21$
• $y^2=101x^6+67x^5+11x^4+123x^3+102x^2+73x+105$
• $y^2=70x^6+77x^5+120x^4+77x^3+85x^2+63x+97$
• $y^2=92x^6+33x^5+134x^4+8x^3+134x^2+33x+92$
• $y^2=112x^6+131x^5+99x^4+41x^3+99x^2+131x+112$
• $y^2=9x^6+28x^5+84x^4+54x^3+79x^2+32x+29$
• $y^2=61x^6+80x^4+82x^3+80x^2+61$
• $y^2=47x^6+29x^5+5x^4+65x^3+5x^2+29x+47$
• $y^2=61x^6+42x^5+130x^4+66x^3+113x^2+137x+105$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{139}$.

Endomorphism algebra over $\F_{139}$

The isogeny class factors as 1.139.ax $\times$ 1.139.at and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.139.ax : \(\Q(\sqrt{-3}) \).
• 1.139.at : \(\Q(\sqrt{-195}) \).

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
2.139.ae_agd	$2$	(not in LMFDB)
2.139.e_agd	$2$	(not in LMFDB)
2.139.bq_bbn	$2$	(not in LMFDB)
2.139.am_fp	$3$	(not in LMFDB)
2.139.ad_aba	$3$	(not in LMFDB)