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

• Label: 2.199.aby_bnf
• 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:	$( 1 - 27 x + 199 x^{2} )( 1 - 23 x + 199 x^{2} )$
	$1 - 50 x + 1019 x^{2} - 9950 x^{3} + 39601 x^{4}$
Frobenius angles:	$\pm0.0936959350875$, $\pm0.196619630811$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$30621$	$1550065641$	$62088089585904$	$2459423877760491849$	$97393991078214868601901$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$150$	$39140$	$7878600$	$1568270884$	$312080606250$	$62103856973006$	$12358664467959750$	$2459374193055320644$	$489415464124909579800$	$97393677359686357343300$

Jacobians and polarizations

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

• $y^2=56x^6+155x^5+138x^4+156x^3+38x^2+125x+119$
• $y^2=67x^6+183x^5+3x^4+23x^3+3x^2+183x+67$
• $y^2=78x^6+59x^5+187x^4+96x^3+28x^2+192x+71$
• $y^2=195x^6+194x^5+114x^4+183x^3+173x^2+x+85$
• $y^2=135x^6+87x^5+120x^4+165x^3+70x^2+13x+33$
• $y^2=19x^6+91x^5+134x^4+4x^3+134x^2+91x+19$
• $y^2=22x^6+7x^5+155x^4+78x^3+155x^2+7x+22$
• $y^2=20x^6+62x^5+x^4+60x^3+x^2+62x+20$
• $y^2=160x^6+118x^5+173x^4+122x^3+183x^2+22x+19$
• $y^2=141x^6+144x^5+26x^4+25x^3+26x^2+144x+141$
• $y^2=59x^6+30x^5+150x^4+53x^3+150x^2+30x+59$
• $y^2=4x^6+198x^5+64x^4+168x^3+154x^2+117x+1$

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.ax 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.ax : \(\Q(\sqrt{-267}) \).

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.ae_aip	$2$	(not in LMFDB)
2.199.e_aip	$2$	(not in LMFDB)
2.199.by_bnf	$2$	(not in LMFDB)