Abelian variety isogeny class 2.131.abq_bba over $\F_{131}$

• Label: 2.131.abq_bba
• Base field: $\F_{131}$
• 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:	$\F_{131}$
Dimension:	$2$
L-polynomial:	$( 1 - 22 x + 131 x^{2} )( 1 - 20 x + 131 x^{2} )$
	$1 - 42 x + 702 x^{2} - 5502 x^{3} + 17161 x^{4}$
Frobenius angles:	$\pm0.0891048534084$, $\pm0.161711276416$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$12320$	$288386560$	$5049102507680$	$86730296891392000$	$1488386936638787108000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$90$	$16802$	$2245950$	$294500238$	$38579746650$	$5053918109042$	$662062688505630$	$86730204189727198$	$11361656660859282330$	$1488377021776546972802$

Jacobians and polarizations

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

• $y^2=28x^6+31x^5+108x^4+100x^3+101x^2+78x+41$
• $y^2=33x^6+89x^5+38x^4+83x^3+38x^2+89x+33$
• $y^2=57x^6+27x^5+57x^4+31x^3+57x^2+27x+57$
• $y^2=125x^6+83x^5+88x^4+110x^3+78x^2+24x+89$
• $y^2=54x^6+118x^5+50x^4+16x^3+31x^2+79x+92$
• $y^2=6x^6+53x^5+113x^4+108x^3+113x^2+53x+6$
• $y^2=72x^6+37x^5+48x^4+84x^3+48x^2+37x+72$
• $y^2=85x^6+2x^5+90x^4+28x^3+71x^2+30x+67$
• $y^2=49x^6+5x^5+54x^4+51x^3+54x^2+5x+49$
• $y^2=67x^5+60x^4+61x^3+39x^2+83x$
• $y^2=69x^6+4x^5+94x^4+11x^3+99x^2+45x+73$
• $y^2=126x^6+42x^5+121x^4+39x^3+121x^2+42x+126$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{131}$.

Endomorphism algebra over $\F_{131}$

The isogeny class factors as 1.131.aw $\times$ 1.131.au and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.131.aw : \(\Q(\sqrt{-10}) \).
• 1.131.au : \(\Q(\sqrt{-31}) \).

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
2.131.ac_agw	$2$	(not in LMFDB)
2.131.c_agw	$2$	(not in LMFDB)
2.131.bq_bba	$2$	(not in LMFDB)