Abelian variety isogeny class 2.157.abs_beo over $\F_{157}$

• Label: 2.157.abs_beo
• Base field: $\F_{157}$
• 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:	$\F_{157}$
Dimension:	$2$
L-polynomial:	$( 1 - 24 x + 157 x^{2} )( 1 - 20 x + 157 x^{2} )$
	$1 - 44 x + 794 x^{2} - 6908 x^{3} + 24649 x^{4}$
Frobenius angles:	$\pm0.0929086555916$, $\pm0.205845557398$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$18492$	$599066832$	$14971819109436$	$369158899399041024$	$9099112025167481580252$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$114$	$24302$	$3868794$	$607595758$	$95389538994$	$14976078944222$	$2351243339560026$	$369145194873633886$	$57955795547652591858$	$9099059901037894613582$

Jacobians and polarizations

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

• $y^2=142x^6+148x^5+22x^4+115x^3+5x^2+31x+95$
• $y^2=x^6+25x^5+76x^4+80x^3+133x^2+90x+134$
• $y^2=151x^6+139x^5+111x^4+152x^3+111x^2+139x+151$
• $y^2=24x^6+75x^5+142x^4+68x^3+94x^2+144x+124$
• $y^2=141x^6+66x^5+90x^4+88x^3+25x^2+8x+111$
• $y^2=5x^6+69x^5+61x^4+93x^3+8x^2+43x+152$
• $y^2=121x^6+112x^5+117x^4+72x^3+77x^2+115x+4$
• $y^2=82x^6+90x^5+121x^4+81x^3+59x^2+42x+137$
• $y^2=59x^6+45x^5+84x^4+72x^3+84x^2+45x+59$
• $y^2=75x^6+24x^5+62x^4+56x^3+62x^2+24x+75$
• $y^2=133x^6+144x^5+16x^4+94x^3+16x^2+144x+133$
• $y^2=27x^6+47x^5+148x^4+22x^3+51x^2+34x+5$
• $y^2=20x^6+103x^5+141x^4+115x^3+76x^2+44x+66$
• $y^2=66x^6+68x^5+9x^4+125x^3+9x^2+68x+66$
• $y^2=65x^6+62x^5+14x^4+98x^3+14x^2+62x+65$
• $y^2=129x^6+68x^5+21x^4+137x^3+21x^2+68x+129$
• $y^2=103x^6+93x^5+105x^4+22x^3+6x^2+99x+120$
• $y^2=17x^6+87x^5+72x^4+127x^3+79x^2+125x+17$
• $y^2=21x^6+3x^5+58x^3+3x+21$
• $y^2=95x^6+144x^5+78x^4+18x^3+119x^2+69x+84$
• and

• $y^2=119x^6+52x^5+52x^4+145x^3+81x^2+83x+129$
• $y^2=147x^6+31x^5+55x^4+142x^3+127x^2+98x+34$
• $y^2=66x^6+127x^5+125x^4+139x^3+77x^2+48x+50$
• $y^2=104x^6+78x^5+89x^4+86x^3+111x^2+5x+63$
• $y^2=156x^6+20x^5+45x^4+111x^3+81x^2+13x+148$
• $y^2=87x^6+147x^5+87x^4+45x^3+87x^2+147x+87$
• $y^2=94x^6+73x^5+29x^4+104x^3+119x^2+71x+155$
• $y^2=52x^6+144x^5+21x^4+28x^3+32x^2+22x+148$
• $y^2=14x^6+132x^5+27x^4+85x^3+74x^2+80x+120$
• $y^2=22x^6+47x^5+142x^4+115x^3+53x^2+106x+77$
• $y^2=32x^6+66x^5+81x^4+57x^3+138x^2+156x+107$
• $y^2=35x^6+53x^5+148x^4+104x^3+129x^2+68x+86$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{157}$.

Endomorphism algebra over $\F_{157}$

The isogeny class factors as 1.157.ay $\times$ 1.157.au and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.157.ay : \(\Q(\sqrt{-13}) \).
• 1.157.au : \(\Q(\sqrt{-57}) \).

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
2.157.ae_agk	$2$	(not in LMFDB)
2.157.e_agk	$2$	(not in LMFDB)
2.157.bs_beo	$2$	(not in LMFDB)