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

• Label: 2.41.ac_ct
• 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:	$1 - 2 x + 71 x^{2} - 82 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.386661326622$, $\pm0.561631648131$
Angle rank:	$2$ (numerical)
Number field:	4.0.3276432.1
Galois group:	$D_{4}$
Jacobians:	$36$
Isomorphism classes:	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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1669$	$3065953$	$4761964096$	$7976754305113$	$13422274294135069$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$40$	$1820$	$69094$	$2822868$	$115852880$	$4750085198$	$194753945216$	$7984930516324$	$327381972326326$	$13422659015688140$

Jacobians and polarizations

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

• $y^2=18 x^6+16 x^5+33 x^4+27 x^3+14 x^2+34 x+14$
• $y^2=33 x^6+23 x^5+28 x^4+28 x^3+27 x^2+29 x+8$
• $y^2=15 x^6+31 x^5+17 x^4+4 x^3+19 x^2+21 x+30$
• $y^2=22 x^6+29 x^5+2 x^4+26 x^3+16 x^2+19 x+5$
• $y^2=40 x^6+23 x^5+23 x^3+33 x^2+13 x+37$
• $y^2=34 x^6+20 x^5+24 x^4+39 x^3+16 x^2+25 x+29$
• $y^2=15 x^6+4 x^5+20 x^4+9 x^3+14 x^2+27 x+24$
• $y^2=17 x^6+24 x^5+29 x^4+7 x^3+8 x^2+40 x+14$
• $y^2=28 x^6+28 x^5+10 x^4+25 x^3+30 x^2+24 x+30$
• $y^2=12 x^6+x^5+21 x^4+25 x^3+21 x^2+38 x+13$
• $y^2=20 x^6+22 x^5+32 x^4+6 x^3+20 x^2+19 x+30$
• $y^2=31 x^6+25 x^5+23 x^4+25 x^3+15 x^2+10 x+24$
• $y^2=3 x^6+37 x^5+8 x^4+33 x^3+27 x^2+40 x+25$
• $y^2=2 x^6+2 x^5+36 x^4+27 x^3+34 x^2+37 x+24$
• $y^2=12 x^6+16 x^5+13 x^4+30 x^3+24 x^2+x+34$
• $y^2=5 x^6+3 x^5+9 x^4+9 x^3+9 x^2+13 x+30$
• $y^2=22 x^6+19 x^5+24 x^4+28 x^3+28 x^2+18 x+6$
• $y^2=18 x^6+11 x^5+x^4+33 x^3+4 x^2+2 x+27$
• $y^2=2 x^6+18 x^5+8 x^4+31 x^3+8 x^2+29 x+36$
• $y^2=x^6+40 x^5+19 x^4+39 x^3+18 x^2+2 x+5$
• and

• $y^2=13 x^6+39 x^5+13 x^4+15 x^3+27 x^2+25 x+39$
• $y^2=17 x^6+29 x^5+4 x^4+x^3+17 x^2+18 x+40$
• $y^2=24 x^6+2 x^5+25 x^4+38 x^3+3 x^2+35 x+19$
• $y^2=40 x^6+9 x^5+16 x^4+35 x^3+25 x^2+33 x+1$
• $y^2=29 x^6+11 x^5+12 x^4+36 x^3+8 x^2+4 x+30$
• $y^2=34 x^6+10 x^5+30 x^4+15 x^3+8 x^2+36 x+26$
• $y^2=12 x^6+3 x^5+39 x^4+31 x^3+35 x^2+29 x+7$
• $y^2=28 x^6+40 x^5+11 x^4+18 x^3+15 x^2+8 x+36$
• $y^2=33 x^6+24 x^5+9 x^4+14 x^3+19 x^2+32 x+30$
• $y^2=11 x^6+34 x^5+29 x^4+36 x^3+3 x^2+23 x+20$
• $y^2=25 x^6+8 x^5+29 x^4+9 x^3+22 x^2+33 x+22$
• $y^2=30 x^6+5 x^5+3 x^4+2 x^2+36 x+34$
• $y^2=5 x^6+4 x^5+17 x^4+6 x^3+4 x^2+13 x+26$
• $y^2=28 x^6+23 x^5+10 x^4+21 x^3+5 x^2+x+4$
• $y^2=18 x^6+14 x^5+21 x^4+38 x^3+17 x^2+8 x+24$
• $y^2=7 x^6+25 x^5+20 x^4+3 x^3+30 x^2+5 x+32$

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 4.0.3276432.1.

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.c_ct	$2$	(not in LMFDB)