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

• Label: 2.41.au_gu
• 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:	$1 - 20 x + 176 x^{2} - 820 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.0753103869890$, $\pm0.299287056274$
Angle rank:	$2$ (numerical)
Number field:	4.0.555264.3
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1018$	$2746564$	$4756935850$	$7987106988304$	$13421744275906378$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$1634$	$69022$	$2826534$	$115848302$	$4749973058$	$194753505542$	$7984925174974$	$327381974247862$	$13422659707241954$

Jacobians and polarizations

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

• $y^2=25x^6+15x^5+29x^4+4x^3+25x^2+9x+9$
• $y^2=19x^6+25x^5+15x^4+32x^3+x^2+19x+3$
• $y^2=3x^6+16x^5+36x^4+27x^3+10x^2+25x+32$
• $y^2=7x^6+13x^5+26x^4+31x^3+31x^2+9x+37$

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 4.0.555264.3.

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.u_gu	$2$	(not in LMFDB)