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

• Label: 2.199.abx_bmj
• Base field: $\F_{199}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{199}$
Dimension:	$2$
L-polynomial:	$1 - 49 x + 997 x^{2} - 9751 x^{3} + 39601 x^{4}$
Frobenius angles:	$\pm0.137603390886$, $\pm0.189051491022$
Angle rank:	$2$ (numerical)
Number field:	4.0.870725.2
Galois group:	$D_{4}$
Jacobians:	$10$

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$30799$	$1552238801$	$62101151528581$	$2459483308781640029$	$97394215085391547199824$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$151$	$39195$	$7880257$	$1568308779$	$312081324036$	$62103868432311$	$12358664617164919$	$2459374194411287619$	$489415464125301942613$	$97393677359344422272350$

Jacobians and polarizations

This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=149x^6+34x^5+171x^4+106x^3+52x^2+193x+166$
• $y^2=150x^6+142x^5+103x^4+173x^3+174x^2+80x+67$
• $y^2=87x^6+59x^5+29x^4+173x^3+162x^2+x+7$
• $y^2=83x^6+144x^5+38x^4+138x^3+85x^2+114x+37$
• $y^2=46x^6+164x^5+188x^4+8x^3+123x^2+12x+68$
• $y^2=106x^6+197x^5+169x^4+74x^3+99x^2+36x+75$
• $y^2=135x^6+25x^5+75x^4+106x^3+87x^2+194x+146$
• $y^2=190x^6+57x^5+59x^4+17x^3+56x^2+140x+83$
• $y^2=50x^6+61x^5+114x^4+26x^3+115x^2+103x+12$
• $y^2=126x^6+102x^5+116x^4+90x^3+73x^2+153x+184$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{199}$

The endomorphism algebra of this simple isogeny class is 4.0.870725.2.

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.bx_bmj	$2$	(not in LMFDB)