Abelian variety isogeny class 2.83.abg_qc over $\F_{83}$

• Label: 2.83.abg_qc
• Base field: $\F_{83}$
• 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:	$\F_{83}$
Dimension:	$2$
L-polynomial:	$( 1 - 18 x + 83 x^{2} )( 1 - 14 x + 83 x^{2} )$
	$1 - 32 x + 418 x^{2} - 2656 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.0496118990883$, $\pm0.221078141621$
Angle rank:	$2$ (numerical)
Jacobians:	$16$
Isomorphism classes:	40

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4620$	$46181520$	$326592868140$	$2252372482483200$	$15516157023790985100$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$52$	$6702$	$571180$	$47460014$	$3939070052$	$326940287454$	$27136044630236$	$2252292131002846$	$186940254258623380$	$15516041180896768782$

Jacobians and polarizations

This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=13x^6+52x^5+48x^4+73x^3+3x^2+29x+19$
• $y^2=22x^6+42x^5+60x^4+33x^3+2x^2+67x+54$
• $y^2=47x^6+52x^5+35x^4+4x^3+62x^2+19x+42$
• $y^2=80x^6+48x^5+79x^4+75x^3+31x^2+29x+35$
• $y^2=45x^6+22x^5+4x^4+50x^3+38x^2+35x+18$
• $y^2=43x^6+72x^5+18x^4+70x^3+67x^2+20x+15$
• $y^2=18x^6+28x^5+7x^4+10x^3+7x^2+28x+18$
• $y^2=80x^6+38x^5+20x^4+8x^3+20x^2+38x+80$
• $y^2=64x^6+65x^5+x^4+58x^3+82x^2+46x+56$
• $y^2=48x^6+54x^5+64x^4+64x^3+64x^2+54x+48$
• $y^2=76x^6+15x^5+62x^4+60x^3+6x^2+34x+15$
• $y^2=39x^6+34x^5+65x^4+44x^3+36x^2+58x+28$
• $y^2=46x^6+65x^5+18x^4+64x^3+18x^2+65x+46$
• $y^2=37x^6+24x^5+22x^4+20x^3+45x^2+70x+56$
• $y^2=67x^6+34x^5+9x^4+67x^3+72x^2+47x+45$
• $y^2=55x^6+67x^5+66x^4+9x^3+52x^2+14x+52$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{83}$.

Endomorphism algebra over $\F_{83}$

The isogeny class factors as 1.83.as $\times$ 1.83.ao and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.83.as : \(\Q(\sqrt{-2}) \).
• 1.83.ao : \(\Q(\sqrt{-34}) \).

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
2.83.ae_adi	$2$	(not in LMFDB)
2.83.e_adi	$2$	(not in LMFDB)
2.83.bg_qc	$2$	(not in LMFDB)