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

• Label: 2.41.ao_eu
• 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:	$1 - 14 x + 124 x^{2} - 574 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.228506026664$, $\pm0.389569638973$
Angle rank:	$2$ (numerical)
Number field:	4.0.8068928.1
Galois group:	$D_{4}$
Jacobians:	$36$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1218$	$2915892$	$4801513122$	$7992354999888$	$13422446649446898$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$28$	$1734$	$69664$	$2828390$	$115854368$	$4750091190$	$194754579740$	$7984925659198$	$327381898079740$	$13422658945926774$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 36 curves (of which all are hyperelliptic):

• $y^2=12 x^6+36 x^5+26 x^4+37 x^3+30 x^2+18 x+24$
• $y^2=24 x^6+40 x^5+8 x^4+3 x^3+17 x^2+40 x+17$
• $y^2=25 x^6+13 x^5+5 x^4+20 x^3+4 x^2+23 x+4$
• $y^2=28 x^6+11 x^5+22 x^4+5 x^3+12 x^2+36 x+24$
• $y^2=4 x^6+25 x^5+18 x^4+39 x^3+31 x^2+16 x+18$
• $y^2=35 x^6+12 x^5+11 x^4+38 x^3+36 x^2+34 x+2$
• $y^2=33 x^6+5 x^4+15 x^3+19 x^2+23 x+4$
• $y^2=26 x^6+32 x^5+38 x^4+18 x^3+38 x^2+3 x+17$
• $y^2=33 x^6+31 x^5+21 x^4+28 x^3+15 x^2+21 x+13$
• $y^2=26 x^6+28 x^5+34 x^4+9 x^3+32 x^2+17 x+6$
• $y^2=17 x^6+26 x^5+11 x^4+32 x^3+13 x^2+34 x+19$
• $y^2=29 x^6+22 x^5+10 x^4+18 x^3+39 x^2+12 x+34$
• $y^2=16 x^6+37 x^5+14 x^4+10 x^3+28 x^2+12$
• $y^2=28 x^6+19 x^5+38 x^4+4 x^3+35 x^2+4 x+32$
• $y^2=31 x^6+10 x^5+13 x^4+24 x^3+38 x^2+21 x+20$
• $y^2=17 x^6+9 x^5+11 x^4+26 x^3+2 x^2+31 x+18$
• $y^2=40 x^5+28 x^4+25 x^3+33 x^2+10 x+6$
• $y^2=38 x^6+15 x^5+17 x^4+12 x^3+11 x^2+38 x+19$
• $y^2=9 x^6+10 x^5+26 x^4+38 x^3+30 x^2+11 x+31$
• $y^2=3 x^6+8 x^5+16 x^4+15 x^3+35 x^2+7 x+36$
• and

• $y^2=11 x^6+38 x^5+16 x^4+39 x^3+31 x^2+14 x+10$
• $y^2=15 x^6+36 x^5+2 x^4+39 x^3+4 x^2+13 x+19$
• $y^2=22 x^6+30 x^5+22 x^4+9 x^3+12 x^2+14 x+28$
• $y^2=26 x^6+20 x^5+3 x^4+21 x^3+28 x^2+27 x+28$
• $y^2=14 x^6+26 x^5+29 x^4+14 x^3+3 x^2+16 x+18$
• $y^2=36 x^6+11 x^5+16 x^4+24 x^3+32 x^2+20 x+6$
• $y^2=30 x^6+14 x^5+26 x^4+23 x^3+38 x^2+21 x+9$
• $y^2=17 x^6+14 x^5+7 x^4+21 x^3+14 x^2+12 x+24$
• $y^2=13 x^6+3 x^5+37 x^3+19 x^2+36 x+29$
• $y^2=28 x^6+9 x^5+33 x^4+21 x^3+39 x^2+8 x+11$
• $y^2=30 x^6+27 x^5+10 x^4+20 x^3+26 x^2+9 x+13$
• $y^2=37 x^6+9 x^5+35 x^4+21 x^3+17 x^2+7 x+26$
• $y^2=x^6+18 x^5+27 x^4+13 x^3+10 x^2+32 x+6$
• $y^2=13 x^6+18 x^5+12 x^4+40 x^3+30 x^2+27 x+24$
• $y^2=40 x^6+28 x^5+26 x^4+23 x^3+30 x^2+11$
• $y^2=34 x^6+18 x^5+26 x^4+10 x^3+29 x^2+27 x+32$

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 4.0.8068928.1.

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.o_eu	$2$	(not in LMFDB)