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

• Label: 2.83.abf_po
• 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:	$( 1 - 17 x + 83 x^{2} )( 1 - 14 x + 83 x^{2} )$
	$1 - 31 x + 404 x^{2} - 2573 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.117184483028$, $\pm0.221078141621$
Angle rank:	$2$ (numerical)
Jacobians:	$8$
Isomorphism classes:	20

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4690$	$46421620$	$326976463240$	$2252839361735200$	$15516640638323543950$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$53$	$6737$	$571850$	$47469849$	$3939192823$	$326941647554$	$27136058240869$	$2252292254304241$	$186940255255274510$	$15516041187750459857$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=26x^6+3x^5+17x^4+43x^3+76x^2+43x+42$
• $y^2=5x^6+48x^5+46x^4+14x^3+49x^2+54x+72$
• $y^2=28x^5+31x^4+60x^3+43x^2+81x+58$
• $y^2=29x^6+79x^5+24x^4+70x^3+63x^2+19x+2$
• $y^2=35x^6+50x^5+12x^4+62x^3+32x^2+49x+24$
• $y^2=57x^6+20x^5+64x^4+64x^3+23x^2+77x+43$
• $y^2=72x^6+9x^5+78x^4+x^3+40x^2+80x+23$
• $y^2=67x^6+20x^5+74x^4+21x^3+37x^2+75x+41$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{83}$

The isogeny class factors as 1.83.ar $\times$ 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.ar : \(\Q(\sqrt{-43}) \).
• 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.ad_acu	$2$	(not in LMFDB)
2.83.d_acu	$2$	(not in LMFDB)
2.83.bf_po	$2$	(not in LMFDB)