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

• Label: 2.83.abf_pp
• 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:	$1 - 31 x + 405 x^{2} - 2573 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.134511996275$, $\pm0.210436945261$
Angle rank:	$2$ (numerical)
Number field:	4.0.174725.2
Galois group:	$D_{4}$
Jacobians:	$6$
Isomorphism classes:	6

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4691$	$46436209$	$327029870429$	$2252945041254701$	$15516784121925902416$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$53$	$6739$	$571943$	$47472075$	$3939229248$	$326942089183$	$27136062113885$	$2252292272620419$	$186940255132632899$	$15516041183431240614$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=79x^6+79x^5+30x^4+43x^3+32x^2+34x+67$
• $y^2=58x^6+x^5+8x^4+9x^3+18x^2+12x+50$
• $y^2=15x^6+67x^5+63x^4+2x^3+58x^2+5x+73$
• $y^2=73x^6+69x^5+39x^4+42x^3+58x^2+38x+3$
• $y^2=15x^6+78x^5+6x^4+76x^3+53x^2+54x+30$
• $y^2=47x^6+58x^5+14x^4+35x^3+8x^2+5x+13$

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 4.0.174725.2.

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.bf_pp	$2$	(not in LMFDB)