Abelian variety isogeny class 2.139.abs_bdg over $\F_{139}$

• Label: 2.139.abs_bdg
• Base field: $\F_{139}$
• 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:	$\F_{139}$
Dimension:	$2$
L-polynomial:	$1 - 44 x + 760 x^{2} - 6116 x^{3} + 19321 x^{4}$
Frobenius angles:	$\pm0.0377284713884$, $\pm0.162150612510$
Angle rank:	$2$ (numerical)
Number field:	4.0.65792.3
Galois group:	$D_{4}$
Jacobians:	$4$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$13922$	$365341124$	$7203929001458$	$139347325722259472$	$2692452179043897658722$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$96$	$18906$	$2682408$	$373284054$	$51888844216$	$7212550695690$	$1002544383388896$	$139353667220446110$	$19370159739293638944$	$2692452204119256352186$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=115x^6+117x^5+115x^4+51x^3+17x^2+51x+33$
• $y^2=137x^6+9x^5+117x^4+126x^3+103x^2+77x+8$
• $y^2=7x^6+133x^5+72x^4+85x^3+54x^2+107x+72$
• $y^2=15x^6+118x^5+39x^4+119x^3+61x^2+8x+43$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{139}$.

Endomorphism algebra over $\F_{139}$

The endomorphism algebra of this simple isogeny class is 4.0.65792.3.

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
2.139.bs_bdg	$2$	(not in LMFDB)