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

• Label: 2.83.abc_ni
• 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:	$( 1 - 18 x + 83 x^{2} )( 1 - 10 x + 83 x^{2} )$
	$1 - 28 x + 346 x^{2} - 2324 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.0496118990883$, $\pm0.315076740302$
Angle rank:	$2$ (numerical)
Jacobians:	$16$
Isomorphism classes:	68

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4884$	$46827792$	$327019555764$	$2252208505181184$	$15515573444515367124$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$56$	$6798$	$571928$	$47456558$	$3938921896$	$326938617918$	$27136037859784$	$2252292208071646$	$186940255874963864$	$15516041194120580718$

Jacobians and polarizations

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

• $y^2=35x^6+15x^5+76x^4+24x^3+76x^2+15x+35$
• $y^2=42x^6+50x^5+37x^4+30x^3+5x^2+48x+72$
• $y^2=66x^6+46x^5+69x^4+65x^3+3x^2+20x+12$
• $y^2=2x^6+14x^5+43x^4+24x^3+31x^2+14x+72$
• $y^2=6x^6+23x^5+79x^4+38x^3+73x^2+64x+79$
• $y^2=53x^6+13x^5+45x^4+39x^3+71x^2+59x+71$
• $y^2=55x^6+48x^5+14x^4+4x^3+37x^2+24x+42$
• $y^2=42x^6+44x^5+5x^4+26x^3+5x^2+44x+42$
• $y^2=42x^6+37x^5+60x^4+45x^3+77x^2+16x+35$
• $y^2=65x^6+32x^5+78x^4+78x^3+6x^2+21x+2$
• $y^2=8x^6+16x^5+71x^4+21x^3+16x^2+47x+42$
• $y^2=2x^6+71x^5+81x^4+4x^3+72x^2+57x+6$
• $y^2=50x^6+65x^5+47x^4+8x^3+18x^2+37x+56$
• $y^2=34x^6+35x^5+77x^4+43x^3+77x^2+35x+34$
• $y^2=47x^6+67x^5+61x^4+62x^3+54x^2+29x+57$
• $y^2=8x^6+54x^5+78x^4+20x^3+25x^2+40x+9$

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.ak 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.ak : \(\Q(\sqrt{-58}) \).

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.ai_ao	$2$	(not in LMFDB)
2.83.i_ao	$2$	(not in LMFDB)
2.83.bc_ni	$2$	(not in LMFDB)