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

• Label: 2.157.abr_bdk
• 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:	$( 1 - 25 x + 157 x^{2} )( 1 - 18 x + 157 x^{2} )$
	$1 - 43 x + 764 x^{2} - 6751 x^{3} + 24649 x^{4}$
Frobenius angles:	$\pm0.0220179720414$, $\pm0.244930514047$
Angle rank:	$2$ (numerical)
Jacobians:	$10$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$18620$	$599712960$	$14971410031760$	$369146271897849600$	$9099049314400407049100$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$115$	$24329$	$3868690$	$607574977$	$95388881575$	$14976065487206$	$2351243131186435$	$369145192334295073$	$57955795523458575610$	$9099059900862217602689$

Jacobians and polarizations

This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=114x^6+129x^5+22x^4+31x^3+104x^2+132x+86$
• $y^2=44x^6+12x^5+132x^4+112x^3+2x^2+150x+88$
• $y^2=38x^6+84x^5+121x^4+122x^3+133x^2+70x+131$
• $y^2=150x^6+34x^5+106x^4+66x^3+147x^2+138x+62$
• $y^2=18x^6+68x^5+41x^4+40x^3+134x^2+47x+97$
• $y^2=86x^6+10x^5+19x^4+72x^3+39x^2+95x+93$
• $y^2=40x^6+38x^5+138x^4+111x^3+101x^2+23x+10$
• $y^2=18x^6+13x^5+87x^4+4x^3+31x^2+16x+84$
• $y^2=107x^6+115x^5+128x^4+13x^3+88x^2+116x+127$
• $y^2=117x^6+41x^5+49x^4+29x^3+130x^2+66x+106$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{157}$

The isogeny class factors as 1.157.az $\times$ 1.157.as 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.az : \(\Q(\sqrt{-3}) \).
• 1.157.as : \(\Q(\sqrt{-19}) \).

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
2.157.ah_afg	$2$	(not in LMFDB)
2.157.h_afg	$2$	(not in LMFDB)
2.157.br_bdk	$2$	(not in LMFDB)
2.157.ah_em	$3$	(not in LMFDB)
2.157.ae_ck	$3$	(not in LMFDB)