Abelian variety isogeny class 2.41.at_gg over $\F_{41}$

• Label: 2.41.at_gg
• Base field: $\F_{41}$
• 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:	$\F_{41}$
Dimension:	$2$
L-polynomial:	$1 - 19 x + 162 x^{2} - 779 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.0407282264755$, $\pm0.336329841408$
Angle rank:	$2$ (numerical)
Number field:	4.0.558092.1
Galois group:	$D_{4}$
Jacobians:	$4$
Isomorphism classes:	4

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1046$	$2763532$	$4752605600$	$7981080416000$	$13419311614315606$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$23$	$1645$	$68960$	$2824401$	$115827303$	$4749867442$	$194753332223$	$7984925474721$	$327381955721120$	$13422659340089805$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=38x^6+35x^5+3x^4+33x^3+12x^2+29x+26$
• $y^2=27x^6+24x^5+2x^4+31x^3+32x^2+27x+35$
• $y^2=14x^6+26x^5+23x^4+22x^3+30x^2+35x+11$
• $y^2=3x^6+9x^5+34x^4+5x^3+23x^2+31x+31$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{41}$.

Endomorphism algebra over $\F_{41}$

The endomorphism algebra of this simple isogeny class is 4.0.558092.1.

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
2.41.t_gg	$2$	(not in LMFDB)