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

• Label: 2.199.abx_bme
• 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:	$( 1 - 27 x + 199 x^{2} )( 1 - 22 x + 199 x^{2} )$
	$1 - 49 x + 992 x^{2} - 9751 x^{3} + 39601 x^{4}$
Frobenius angles:	$\pm0.0936959350875$, $\pm0.215336333813$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$30794$	$1551832836$	$62095352213096$	$2459439191319843744$	$97393981500807383327774$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$151$	$39185$	$7879522$	$1568280649$	$312080575561$	$62103853238906$	$12358664378601559$	$2459374191748304689$	$489415464114650116078$	$97393677359753895392825$

Jacobians and polarizations

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

• $y^2=63x^6+8x^5+117x^4+62x^3+62x^2+75x+177$
• $y^2=177x^6+35x^5+43x^4+170x^3+139x^2+48x+6$
• $y^2=147x^6+62x^5+176x^4+85x^3+146x^2+114x+25$
• $y^2=48x^6+13x^5+170x^4+160x^3+183x^2+69x+163$
• $y^2=178x^6+94x^5+72x^4+172x^3+82x^2+89x+120$
• $y^2=27x^6+158x^5+119x^4+60x^3+146x^2+196x+136$
• $y^2=119x^6+63x^5+151x^4+64x^3+31x^2+163x$
• $y^2=37x^6+120x^5+118x^4+169x^3+128x^2+30x+182$
• $y^2=186x^6+12x^5+193x^4+85x^3+67x^2+137x+81$
• $y^2=61x^6+130x^5+124x^4+51x^3+11x^2+67x+74$
• $y^2=62x^6+160x^5+87x^4+184x^3+7x^2+86x+116$
• $y^2=134x^6+77x^5+142x^4+7x^3+147x^2+187x+121$

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.aw 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.aw : \(\Q(\sqrt{-78}) \).

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.af_aho	$2$	(not in LMFDB)
2.199.f_aho	$2$	(not in LMFDB)
2.199.bx_bme	$2$	(not in LMFDB)