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

• Label: 2.83.a_agc
• 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:	$( 1 - 18 x + 83 x^{2} )( 1 + 18 x + 83 x^{2} )$
	$1 - 158 x^{2} + 6889 x^{4}$
Frobenius angles:	$\pm0.0496118990883$, $\pm0.950388100912$
Angle rank:	$1$ (numerical)
Jacobians:	$18$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6732$	$45319824$	$326939694444$	$2251230714602496$	$15516041187109801932$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$6574$	$571788$	$47435950$	$3939040644$	$326939015518$	$27136050989628$	$2252292171719134$	$186940255267540404$	$15516041187013750414$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

• $y^2=3 x^6+14 x^5+50 x^4+51 x^3+9 x^2+47 x+56$
• $y^2=81 x^6+12 x^5+42 x^4+33 x^3+40 x^2+25 x+54$
• $y^2=13 x^6+70 x^5+59 x^4+78 x^3+52 x^2+75 x+38$
• $y^2=51 x^6+76 x^5+81 x^4+46 x^3+15 x^2+42 x+54$
• $y^2=68 x^6+53 x^5+10 x^4+24 x^3+56 x^2+22 x+45$
• $y^2=76 x^6+11 x^5+22 x^4+62 x^3+69 x^2+12 x+81$
• $y^2=11 x^6+48 x^4+23 x^3+82 x^2+2$
• $y^2=x^5+x$
• $y^2=75 x^6+11 x^5+41 x^4+33 x^3+67 x^2+59 x+53$
• $y^2=54 x^6+32 x^5+16 x^4+17 x^3+19 x^2+14 x+30$
• $y^2=59 x^6+x^5+33 x^4+79 x^3+50 x^2+x+24$
• $y^2=4 x^6+23 x^5+53 x^4+11 x^3+37 x^2+36 x+76$
• $y^2=40 x^6+65 x^5+19 x^4+32 x^3+37 x^2+69 x+80$
• $y^2=24 x^6+75 x^5+72 x^4+52 x^3+78 x^2+69 x+4$
• $y^2=41 x^6+2 x^5+33 x^4+15 x^3+73 x^2+20 x+82$
• $y^2=51 x^6+13 x^5+58 x^4+30 x^3+26 x^2+79 x+13$
• $y^2=10 x^6+7 x^5+76 x^4+51 x^3+23 x^2+40 x+46$
• $y^2=10 x^6+72 x^5+8 x^4+23 x^3+48 x^2+19 x+2$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{83^{2}}$.

Endomorphism algebra over $\F_{83}$

The isogeny class factors as 1.83.as $\times$ 1.83.s 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.s : \(\Q(\sqrt{-2}) \).

Endomorphism algebra over $\overline{\F}_{83}$

The base change of $A$ to $\F_{83^{2}}$ is 1.6889.agc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.83.abk_sw	$2$	(not in LMFDB)
2.83.bk_sw	$2$	(not in LMFDB)
2.83.a_gc	$4$	(not in LMFDB)