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

• Label: 2.41.au_gt
• 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:	$1 - 20 x + 175 x^{2} - 820 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.0504478270356$, $\pm0.305285834519$
Angle rank:	$2$ (numerical)
Number field:	4.0.352016.1
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1017$	$2742849$	$4752762372$	$7984567838601$	$13420701599035977$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$1632$	$68962$	$2825636$	$115839302$	$4749905142$	$194753078822$	$7984922524228$	$327381954578482$	$13422659541937152$

Jacobians and polarizations

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

• $y^2=25x^6+22x^5+22x^4+3x^3+36x^2+17x+15$
• $y^2=19x^6+23x^5+x^4+35x^3+7x^2+3x+38$
• $y^2=3x^6+20x^5+7x^4+24x^3+40x^2+4x+1$
• $y^2=5x^6+32x^5+11x^4+13x^3+15x^2+29x+1$

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 4.0.352016.1.

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.u_gt	$2$	(not in LMFDB)