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

• Label: 2.131.abn_yo
• Base field: $\F_{131}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{131}$
Dimension:	$2$
L-polynomial:	$1 - 39 x + 638 x^{2} - 5109 x^{3} + 17161 x^{4}$
Frobenius angles:	$\pm0.109018476986$, $\pm0.224314856348$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{17})\)
Galois group:	$C_2^2$
Jacobians:	$26$

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$12652$	$290338096$	$5053915237072$	$86737773366645696$	$1488393224867431504612$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$93$	$16917$	$2248092$	$294525625$	$38579909643$	$5053917329862$	$662062648500801$	$86730203540650609$	$11361656654439817572$	$1488377021749977601077$

Jacobians and polarizations

This isogeny class contains the Jacobians of 26 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=98x^6+90x^5+89x^4+93x^3+122x^2+100x+31$
• $y^2=10x^6+21x^5+75x^3+39x^2+82x+15$
• $y^2=17x^6+126x^5+85x^4+26x^3+94x^2+110x+108$
• $y^2=121x^6+12x^5+52x^4+23x^3+42x^2+79x+64$
• $y^2=114x^6+102x^5+11x^4+41x^3+92x^2+130x+105$
• $y^2=114x^6+68x^5+57x^4+109x^3+97x^2+97x+23$
• $y^2=73x^6+118x^5+43x^4+99x^3+122x^2+48x+128$
• $y^2=47x^6+12x^5+x^4+52x^3+15x^2+30x+28$
• $y^2=81x^6+91x^5+75x^4+115x^3+74x^2+55x+98$
• $y^2=120x^6+70x^5+130x^4+17x^3+63x^2+16x+18$
• $y^2=104x^6+21x^5+83x^4+54x^3+87x^2+123x+49$
• $y^2=66x^6+82x^5+25x^4+70x^3+111x^2+117x+46$
• $y^2=41x^6+2x^5+6x^4+106x^3+38x^2+116x+86$
• $y^2=117x^6+101x^5+107x^4+93x^3+25x^2+29x+78$
• $y^2=116x^6+117x^5+87x^4+102x^3+61x^2+113x+40$
• $y^2=31x^6+112x^5+110x^4+103x^3+79x^2+44x+71$
• $y^2=18x^6+52x^5+84x^4+56x^3+12x^2+16x+57$
• $y^2=117x^6+94x^5+26x^4+35x^3+130x^2+20x+111$
• $y^2=96x^6+77x^5+101x^4+122x^3+108x^2+106x+96$
• $y^2=128x^6+16x^5+115x^4+80x^3+3x^2+112x+62$
• $y^2=97x^6+7x^5+33x^4+68x^3+84x^2+35x+57$
• $y^2=50x^6+129x^5+84x^4+72x^3+67x^2+38x+98$
• $y^2=x^6+34x^5+106x^4+36x^3+15x^2+35x+92$
• $y^2=44x^6+113x^5+38x^4+110x^3+127x^2+115x+106$
• $y^2=107x^6+100x^5+24x^4+99x^3+5x^2+120x+51$
• $y^2=x^6+6x^5+3x^4+120x^3+38x^2+35x+31$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{131^{6}}$.

Endomorphism algebra over $\F_{131}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{17})\).

Endomorphism algebra over $\overline{\F}_{131}$

The base change of $A$ to $\F_{131^{6}}$ is 1.5053913144281.epbvy^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{131^{2}}$

  The base change of $A$ to $\F_{131^{2}}$ is the simple isogeny class 2.17161.ajl_clkq and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{17})\).

• Endomorphism algebra over $\F_{131^{3}}$

  The base change of $A$ to $\F_{131^{3}}$ is the simple isogeny class 2.2248091.a_epbvy and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{17})\).

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
2.131.bn_yo	$2$	(not in LMFDB)
2.131.a_jl	$3$	(not in LMFDB)
2.131.bn_yo	$3$	(not in LMFDB)