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

• Label: 2.83.abe_pa
• 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:	$( 1 - 16 x + 83 x^{2} )( 1 - 14 x + 83 x^{2} )$
	$1 - 30 x + 390 x^{2} - 2490 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.158801688027$, $\pm0.221078141621$
Angle rank:	$2$ (numerical)
Jacobians:	$24$
Isomorphism classes:	64

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4760$	$46648000$	$327301660280$	$2253173036800000$	$15516903266016567800$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$54$	$6770$	$572418$	$47476878$	$3939259494$	$326942097410$	$27136059223698$	$2252292223889758$	$186940254593976534$	$15516041178842080850$

Jacobians and polarizations

This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=69x^6+38x^5+9x^4+18x^3+9x^2+38x+69$
• $y^2=80x^6+55x^5+61x^4+66x^3+61x^2+55x+80$
• $y^2=6x^6+11x^5+62x^4+19x^3+18x^2+3x+18$
• $y^2=14x^6+60x^5+17x^4+57x^3+17x^2+60x+14$
• $y^2=45x^6+25x^5+54x^4+76x^3+15x^2+36x+18$
• $y^2=24x^6+46x^5+64x^4+20x^3+64x^2+46x+24$
• $y^2=74x^6+69x^5+39x^4+12x^3+39x^2+69x+74$
• $y^2=43x^6+55x^5+32x^4+15x^3+8x^2+19x+72$
• $y^2=76x^6+55x^5+67x^4+26x^3+67x^2+55x+76$
• $y^2=81x^6+46x^5+34x^4+34x^3+34x^2+46x+81$
• $y^2=55x^6+80x^5+34x^4+58x^3+34x^2+80x+55$
• $y^2=82x^6+66x^5+30x^4+15x^3+23x^2+15x+8$
• $y^2=40x^6+78x^5+3x^4+63x^3+3x^2+78x+40$
• $y^2=67x^6+53x^5+68x^4+32x^3+68x^2+53x+67$
• $y^2=63x^6+49x^5+57x^4+69x^3+57x^2+49x+63$
• $y^2=82x^6+72x^5+82x^4+44x^3+73x^2+62x+79$
• $y^2=67x^6+60x^5+29x^4+20x^3+51x^2+32x+72$
• $y^2=6x^6+26x^5+74x^4+71x^3+19x^2+38x+73$
• $y^2=34x^6+73x^5+80x^4+43x^3+71x^2+6x+18$
• $y^2=5x^6+25x^5+74x^4+74x^3+54x^2+70x+82$
• $y^2=17x^6+71x^5+59x^4+75x^3+59x^2+71x+17$
• $y^2=31x^6+45x^5+57x^4+70x^3+50x^2+79x+28$
• $y^2=56x^6+15x^5+59x^4+5x^3+26x^2+47x+46$
• $y^2=45x^6+70x^5+22x^4+59x^3+15x^2+37x+72$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{83}$

The isogeny class factors as 1.83.aq $\times$ 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.aq : \(\Q(\sqrt{-19}) \).
• 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.ac_acg	$2$	(not in LMFDB)
2.83.c_acg	$2$	(not in LMFDB)
2.83.be_pa	$2$	(not in LMFDB)