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

• Label: 2.83.abc_nk
• Base field: $\F_{83}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{83}$
Dimension:	$2$
L-polynomial:	$1 - 28 x + 348 x^{2} - 2324 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.0731644126318$, $\pm0.309647214466$
Angle rank:	$2$ (numerical)
Number field:	4.0.12255488.1
Galois group:	$D_{4}$
Jacobians:	$24$
Isomorphism classes:	24

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4886$	$46856740$	$327115906838$	$2252374440621200$	$15515764249987534246$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$56$	$6802$	$572096$	$47460054$	$3938970336$	$326939124034$	$27136042378760$	$2252292250476126$	$186940256358455768$	$15516041199965196882$

Jacobians and polarizations

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

• $y^2=18x^6+57x^5+30x^4+6x^3+2x^2+66x+1$
• $y^2=8x^6+77x^5+33x^4+64x^2+61x+72$
• $y^2=36x^6+24x^5+50x^4+6x^3+11x^2+28x+54$
• $y^2=46x^6+74x^5+52x^4+41x^3+74x^2+42x+10$
• $y^2=81x^6+66x^5+2x^4+52x^3+5x^2+29x+42$
• $y^2=74x^6+3x^5+6x^4+15x^3+22x^2+43x+69$
• $y^2=67x^6+29x^5+73x^4+42x^3+69x^2+27x+72$
• $y^2=79x^6+13x^5+53x^4+2x^3+75x^2+27x+24$
• $y^2=22x^6+81x^5+21x^4+78x^3+71x^2+63x+24$
• $y^2=34x^6+79x^5+73x^4+52x^3+8x^2+78x+44$
• $y^2=43x^6+79x^5+68x^4+79x^3+41x^2+53x+66$
• $y^2=30x^6+78x^5+61x^4+36x^3+78x^2+26x+77$
• $y^2=54x^6+76x^5+30x^4+55x^3+8x^2+21x+52$
• $y^2=66x^6+18x^5+40x^4+15x^3+13x^2+4x+20$
• $y^2=49x^6+65x^5+68x^4+12x^3+76x^2+5x+24$
• $y^2=50x^6+68x^5+30x^4+26x^3+6x^2+73x+47$
• $y^2=35x^6+50x^5+16x^4+48x^3+78x^2+80x+3$
• $y^2=38x^6+30x^5+57x^4+45x^3+67x^2+39x+54$
• $y^2=x^6+54x^5+82x^4+35x^3+8x^2+20x+1$
• $y^2=66x^6+29x^5+74x^4+68x^3+56x^2+57x+80$
• $y^2=19x^6+2x^5+48x^4+19x^3+23x^2+63x+63$
• $y^2=8x^6+24x^5+72x^3+56x^2+8x+67$
• $y^2=68x^6+59x^5+49x^4+60x^3+74x^2+57x+37$
• $y^2=72x^6+13x^5+14x^4+41x^3+40x^2+18x+4$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{83}$

The endomorphism algebra of this simple isogeny class is 4.0.12255488.1.

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.bc_nk	$2$	(not in LMFDB)