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

• Label: 2.139.abp_bau
• Base field: $\F_{139}$
• 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:	$\F_{139}$
Dimension:	$2$
L-polynomial:	$( 1 - 22 x + 139 x^{2} )( 1 - 19 x + 139 x^{2} )$
	$1 - 41 x + 696 x^{2} - 5699 x^{3} + 19321 x^{4}$
Frobenius angles:	$\pm0.117174211439$, $\pm0.201746658314$
Angle rank:	$2$ (numerical)
Jacobians:	$12$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$14278$	$367772724$	$7211452111864$	$139364104961648160$	$2692482135346412639098$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$99$	$19033$	$2685210$	$373329001$	$51889421529$	$7212556850866$	$1002544439162211$	$139353667682240401$	$19370159743729368750$	$2692452204182354269753$

Jacobians and polarizations

This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=20x^6+90x^5+76x^4+2x^3+95x^2+66x+128$
• $y^2=74x^6+13x^5+124x^4+3x^3+124x^2+44x+26$
• $y^2=63x^6+74x^5+93x^4+109x^3+57x^2+111x+72$
• $y^2=78x^6+81x^5+16x^4+133x^3+80x^2+62x+75$
• $y^2=25x^6+68x^5+64x^4+79x^3+39x^2+22x+72$
• $y^2=5x^6+123x^5+118x^4+66x^3+119x^2+18x+100$
• $y^2=91x^6+113x^5+106x^4+37x^3+43x^2+19x+2$
• $y^2=94x^6+104x^5+124x^4+93x^3+78x^2+4x+43$
• $y^2=111x^6+121x^5+114x^4+127x^3+94x^2+35x+94$
• $y^2=108x^6+64x^5+85x^4+35x^3+41x^2+122x+131$
• $y^2=10x^6+18x^5+59x^4+98x^3+111x^2+x+76$
• $y^2=20x^6+17x^5+75x^4+71x^3+11x^2+31x+50$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{139}$

The isogeny class factors as 1.139.aw $\times$ 1.139.at and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.139.aw : \(\Q(\sqrt{-2}) \).
• 1.139.at : \(\Q(\sqrt{-195}) \).

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.ad_afk	$2$	(not in LMFDB)
2.139.d_afk	$2$	(not in LMFDB)
2.139.bp_bau	$2$	(not in LMFDB)