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

• Label: 2.199.ace_btm
• 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:	$( 1 - 28 x + 199 x^{2} )^{2}$
	$1 - 56 x + 1182 x^{2} - 11144 x^{3} + 39601 x^{4}$
Frobenius angles:	$\pm0.0391815390403$, $\pm0.0391815390403$
Angle rank:	$1$ (numerical)
Jacobians:	$1$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$29584$	$1537894656$	$62021358132496$	$2459155292187070464$	$97393107960684543767824$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$144$	$38830$	$7870128$	$1568099614$	$312077776464$	$62103817289806$	$12358663989592176$	$2459374188083670334$	$489415464079550291472$	$97393677359278894394350$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=3x^6+118$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{199}$

The isogeny class factors as 1.199.abc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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
2.199.a_aow	$2$	(not in LMFDB)
2.199.ce_btm	$2$	(not in LMFDB)
2.199.ar_dm	$3$	(not in LMFDB)
2.199.al_ada	$3$	(not in LMFDB)
2.199.w_tz	$3$	(not in LMFDB)
2.199.bc_wn	$3$	(not in LMFDB)
2.199.bi_bal	$3$	(not in LMFDB)