Abelian variety isogeny class 2.127.abq_bat over $\F_{127}$

• Label: 2.127.abq_bat
• Base field: $\F_{127}$
• 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:	$\F_{127}$
Dimension:	$2$
L-polynomial:	$( 1 - 21 x + 127 x^{2} )^{2}$
	$1 - 42 x + 695 x^{2} - 5334 x^{3} + 16129 x^{4}$
Frobenius angles:	$\pm0.118304318667$, $\pm0.118304318667$
Angle rank:	$1$ (numerical)
Jacobians:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$11449$	$254179249$	$4190716671376$	$67673824264408761$	$1091540664935025415009$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$86$	$15756$	$2045864$	$260139220$	$33038575586$	$4195877933022$	$532875939365870$	$67675235266898404$	$8594754760094421368$	$1091533853184292325436$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):

• $y^2=3 x^6+20 x^5+124 x^4+60 x^3+124 x^2+20 x+3$
• $y^2=118 x^6+111 x^5+38 x^4+71 x^3+17 x^2+121 x+99$
• $y^2=63 x^6+63 x^5+126 x^4+119 x^3+14 x^2+93 x+53$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{127}$.

Endomorphism algebra over $\F_{127}$

The isogeny class factors as 1.127.av^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-67}) \)$)$

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
2.127.a_ahf	$2$	(not in LMFDB)
2.127.bq_bat	$2$	(not in LMFDB)
2.127.v_mc	$3$	(not in LMFDB)