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

• Label: 2.41.au_gv
• 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:	$1 - 20 x + 177 x^{2} - 820 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.0953421678685$, $\pm0.292668588087$
Angle rank:	$2$ (numerical)
Number field:	4.0.37025.1
Galois group:	$D_{4}$
Jacobians:	$14$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1019$	$2750281$	$4761110384$	$7989635062025$	$13422763808553979$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$1636$	$69082$	$2827428$	$115857102$	$4750035886$	$194753844622$	$7984926680388$	$327381981833482$	$13422659764981956$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):

• $y^2=15 x^6+17 x^5+20 x^4+38 x^3+34 x^2+23 x+3$
• $y^2=27 x^6+4 x^5+3 x^4+8 x^3+29 x^2+2 x+24$
• $y^2=28 x^6+39 x^5+17 x^4+9 x^3+10 x^2+8 x+26$
• $y^2=6 x^6+17 x^5+37 x^4+5 x^3+23 x^2+39 x+10$
• $y^2=22 x^6+32 x^5+11 x^4+34 x^3+34 x+38$
• $y^2=28 x^6+15 x^5+15 x^4+22 x^3+24 x^2+15 x+17$
• $y^2=30 x^6+29 x^5+29 x^4+9 x^2+14 x+15$
• $y^2=11 x^6+20 x^5+8 x^4+33 x^3+40 x^2+24 x+14$
• $y^2=17 x^6+2 x^5+21 x^4+30 x^3+2 x^2+10 x+15$
• $y^2=11 x^6+22 x^5+16 x^4+6 x^2+24 x+11$
• $y^2=22 x^6+16 x^5+32 x^4+34 x^3+24 x^2+23 x+32$
• $y^2=6 x^6+18 x^5+12 x^4+26 x^3+28 x^2+21 x+34$
• $y^2=6 x^6+34 x^5+38 x^4+12 x^3+39 x^2+25 x+32$
• $y^2=16 x^6+40 x^5+18 x^4+11 x^3+13 x^2+36 x+19$

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 4.0.37025.1.

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_gv	$2$	(not in LMFDB)