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

• Label: 2.157.abr_bdu
• 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:	$( 1 - 23 x + 157 x^{2} )( 1 - 20 x + 157 x^{2} )$
	$1 - 43 x + 774 x^{2} - 6751 x^{3} + 24649 x^{4}$
Frobenius angles:	$\pm0.129963694588$, $\pm0.205845557398$
Angle rank:	$2$ (numerical)
Jacobians:	$12$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$18630$	$600221340$	$14976410487840$	$369172521522057600$	$9099145295250230145150$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$115$	$24349$	$3869980$	$607618177$	$95389887775$	$14976083515066$	$2351243388541435$	$369145195238458273$	$57955795547512592860$	$9099059900964143824789$

Jacobians and polarizations

This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=18x^6+93x^5+116x^4+99x^3+83x^2+71x+71$
• $y^2=136x^6+100x^5+23x^4+91x^3+39x^2+104x+6$
• $y^2=108x^6+107x^5+135x^4+29x^3+45x^2+101x+104$
• $y^2=6x^5+140x^4+28x^3+97x^2+67x+36$
• $y^2=23x^6+44x^5+11x^4+44x^3+91x^2+29x+69$
• $y^2=139x^6+130x^5+143x^4+109x^3+153x^2+57x+84$
• $y^2=148x^6+86x^5+14x^4+78x^3+31x^2+43x+55$
• $y^2=2x^6+59x^5+68x^4+46x^3+104x^2+45x+85$
• $y^2=146x^6+30x^5+47x^4+155x^3+135x^2+29x+64$
• $y^2=10x^6+129x^5+152x^4+3x^3+125x^2+31x+80$
• $y^2=48x^6+90x^5+120x^4+73x^3+68x^2+154x+86$
• $y^2=21x^6+67x^5+92x^4+7x^3+154x^2+95x+152$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{157}$

The isogeny class factors as 1.157.ax $\times$ 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.ax : \(\Q(\sqrt{-11}) \).
• 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.ad_afq	$2$	(not in LMFDB)
2.157.d_afq	$2$	(not in LMFDB)
2.157.br_bdu	$2$	(not in LMFDB)