Abelian variety isogeny class 2.199.abz_bog over $\F_{199}$

• Label: 2.199.abz_bog
• Base field: $\F_{199}$
• 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:	$\F_{199}$
Dimension:	$2$
L-polynomial:	$( 1 - 27 x + 199 x^{2} )( 1 - 24 x + 199 x^{2} )$
	$1 - 51 x + 1046 x^{2} - 10149 x^{3} + 39601 x^{4}$
Frobenius angles:	$\pm0.0936959350875$, $\pm0.176204172288$
Angle rank:	$2$ (numerical)
Jacobians:	$16$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$30448$	$1548219904$	$62079739927744$	$2459401102663464960$	$97393967494055688776848$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$149$	$39093$	$7877540$	$1568256361$	$312080530679$	$62103859165086$	$12358664548923401$	$2459374194646446001$	$489415464146272214540$	$97393677359849276554653$

Jacobians and polarizations

This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=64x^6+79x^5+18x^4+38x^3+105x^2+66x+198$
• $y^2=191x^6+16x^5+177x^4+29x^3+88x^2+188x+85$
• $y^2=186x^6+140x^5+191x^4+48x^3+59x^2+189x+101$
• $y^2=11x^6+117x^5+100x^4+142x^3+163x^2+171x+163$
• $y^2=110x^6+147x^5+17x^4+20x^3+164x^2+24x+99$
• $y^2=142x^6+16x^5+33x^4+25x^3+189x^2+31x+42$
• $y^2=26x^6+121x^5+19x^4+163x^3+32x^2+81x+40$
• $y^2=190x^6+89x^5+53x^4+32x^3+36x^2+104x+31$
• $y^2=107x^6+114x^5+9x^4+108x^3+52x^2+165x+125$
• $y^2=179x^6+111x^5+186x^4+119x^3+140x^2+191x+142$
• $y^2=2x^6+181x^5+128x^4+162x^3+141x^2+85x+37$
• $y^2=189x^6+80x^5+88x^4+138x^3+94x^2+106x+176$
• $y^2=39x^6+6x^5+37x^4+13x^3+139x^2+72x+114$
• $y^2=157x^6+176x^5+20x^4+128x^3+144x^2+36x+3$
• $y^2=198x^6+86x^5+18x^4+37x^3+16x^2+76x+60$
• $y^2=12x^6+62x^5+152x^4+189x^3+161x^2+197x+194$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{199}$.

Endomorphism algebra over $\F_{199}$

The isogeny class factors as 1.199.abb $\times$ 1.199.ay and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.199.abb : \(\Q(\sqrt{-67}) \).
• 1.199.ay : \(\Q(\sqrt{-55}) \).

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
2.199.ad_ajq	$2$	(not in LMFDB)
2.199.d_ajq	$2$	(not in LMFDB)
2.199.bz_bog	$2$	(not in LMFDB)