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

• Label: 2.139.abr_bck
• 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:	$( 1 - 23 x + 139 x^{2} )( 1 - 20 x + 139 x^{2} )$
	$1 - 43 x + 738 x^{2} - 5977 x^{3} + 19321 x^{4}$
Frobenius angles:	$\pm0.0707251543800$, $\pm0.177693164435$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$14040$	$366163200$	$7206548862240$	$139353443025638400$	$2692463915889670408200$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$97$	$18949$	$2683384$	$373300441$	$51889070407$	$7212553404262$	$1002544413084493$	$139353667547084401$	$19370159743384708936$	$2692452204180515360149$

Jacobians and polarizations

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

• $y^2=127x^6+29x^5+37x^4+119x^3+51x^2+62x+48$
• $y^2=130x^6+81x^5+70x^4+10x^3+44x^2+81x+23$
• $y^2=52x^6+120x^5+113x^4+81x^3+30x^2+134x+44$
• $y^2=114x^6+118x^5+64x^4+64x^3+128x^2+80x+16$
• $y^2=72x^6+101x^5+11x^4+93x^3+50x^2+7x+12$
• $y^2=92x^6+37x^5+20x^4+6x^3+3x^2+138x+84$
• $y^2=108x^6+66x^5+75x^4+93x^3+90x^2+65x+129$
• $y^2=103x^6+28x^5+58x^4+17x^3+61x^2+51x+74$
• $y^2=128x^6+29x^5+96x^4+71x^3+5x^2+100x+3$
• $y^2=27x^6+50x^5+136x^4+135x^3+79x^2+81x+130$
• $y^2=95x^6+87x^5+118x^4+6x^3+26x^2+82x+70$
• $y^2=106x^6+133x^5+60x^4+74x^3+57x^2+85x+14$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{139}$

The isogeny class factors as 1.139.ax $\times$ 1.139.au 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.ax : \(\Q(\sqrt{-3}) \).
• 1.139.au : \(\Q(\sqrt{-39}) \).

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
2.139.ad_aha	$2$	(not in LMFDB)
2.139.d_aha	$2$	(not in LMFDB)
2.139.br_bck	$2$	(not in LMFDB)
2.139.an_fi	$3$	(not in LMFDB)
2.139.ae_abq	$3$	(not in LMFDB)