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

• Label: 2.41.c_ap
• 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:	$1 + 2 x - 15 x^{2} + 82 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.255433653965$, $\pm0.824068551096$
Angle rank:	$2$ (numerical)
Number field:	4.0.245312.1
Galois group:	$D_{4}$
Jacobians:	$45$
Isomorphism classes:	45

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1751$	$2771833$	$4773933404$	$8000089351097$	$13421948139018631$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$44$	$1648$	$69266$	$2831124$	$115850064$	$4750225894$	$194752920560$	$7984921250340$	$327381922394210$	$13422659180945568$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 45 curves (of which all are hyperelliptic):

• $y^2=2 x^6+3 x^5+5 x^4+24 x^3+39 x^2+24 x+19$
• $y^2=19 x^6+24 x^5+36 x^4+4 x^3+4 x^2+35 x+17$
• $y^2=20 x^6+27 x^5+39 x^4+x^3+18 x^2+33 x+19$
• $y^2=40 x^6+6 x^5+10 x^4+39 x^3+33 x^2+14 x+33$
• $y^2=23 x^6+31 x^5+23 x^4+5 x^3+27 x^2+39 x+40$
• $y^2=3 x^6+29 x^5+33 x^4+34 x^3+16 x^2+33 x+8$
• $y^2=16 x^6+31 x^5+7 x^4+23 x^3+40 x^2+24 x+16$
• $y^2=25 x^6+27 x^5+23 x^4+26 x^3+35 x^2+21 x+23$
• $y^2=33 x^6+29 x^5+26 x^4+2 x^3+x^2+15 x+6$
• $y^2=4 x^6+20 x^5+15 x^4+35 x^3+35 x^2+36 x+21$
• $y^2=9 x^6+7 x^5+7 x^4+7 x^3+8 x^2+20 x+5$
• $y^2=14 x^6+8 x^5+36 x^4+25 x^3+37 x^2+25 x+15$
• $y^2=15 x^6+2 x^5+3 x^4+10 x^3+6 x^2+7 x+18$
• $y^2=30 x^6+34 x^5+10 x^4+10 x^3+20 x^2+6 x+26$
• $y^2=23 x^6+12 x^5+38 x^4+33 x^3+18 x^2+23 x+39$
• $y^2=40 x^6+34 x^5+20 x^4+27 x^3+4 x^2+14 x+13$
• $y^2=31 x^6+14 x^5+28 x^4+12 x^3+14 x^2+8 x+19$
• $y^2=6 x^6+17 x^5+30 x^4+3 x^3+35 x^2+3 x+26$
• $y^2=37 x^6+3 x^5+27 x^4+34 x^3+30 x^2+8 x+20$
• $y^2=18 x^6+35 x^5+12 x^4+37 x^3+2 x^2+3 x+20$
• and

• $y^2=36 x^6+14 x^5+38 x^4+40 x^3+21 x^2+13 x+22$
• $y^2=20 x^6+22 x^5+15 x^4+39 x^3+29 x^2+13 x+40$
• $y^2=31 x^6+26 x^5+28 x^4+22 x^3+17 x^2+5 x+26$
• $y^2=13 x^6+16 x^5+27 x^4+18 x^3+38 x^2+14 x+34$
• $y^2=18 x^6+26 x^5+38 x^4+24 x^3+21 x^2+19 x+10$
• $y^2=24 x^6+29 x^5+31 x^4+9 x^3+24 x^2+26 x+28$
• $y^2=8 x^6+28 x^5+22 x^4+36 x^3+33 x^2+8 x+39$
• $y^2=27 x^6+28 x^5+27 x^4+8 x^3+24 x^2+33 x+33$
• $y^2=16 x^6+36 x^5+30 x^4+32 x^3+25 x^2+36 x+27$
• $y^2=37 x^6+34 x^5+5 x^4+7 x^3+23 x^2+5 x+4$
• $y^2=33 x^6+18 x^5+16 x^4+21 x^3+29 x^2+7 x+11$
• $y^2=40 x^6+22 x^5+31 x^4+32 x^3+20 x^2+2 x+20$
• $y^2=x^6+11 x^5+9 x^4+7 x^3+18 x^2+29 x+6$
• $y^2=18 x^6+28 x^5+25 x^4+19 x^3+25 x^2+19 x+16$
• $y^2=30 x^6+x^4+22 x^2+x+27$
• $y^2=33 x^6+15 x^5+23 x^4+2 x^3+24 x^2+39 x+34$
• $y^2=10 x^6+7 x^5+22 x^4+7 x^3+16 x^2+13 x+24$
• $y^2=4 x^6+39 x^5+30 x^4+8 x^3+27 x^2+9 x+30$
• $y^2=40 x^6+3 x^5+11 x^4+27 x^3+x^2+31 x+39$
• $y^2=27 x^6+16 x^5+8 x^4+30 x^3+11 x^2+36 x+35$
• $y^2=36 x^6+32 x^5+19 x^4+20 x^3+2 x^2+5 x+25$
• $y^2=17 x^6+2 x^5+3 x^4+36 x^3+27 x^2+13 x+2$
• $y^2=12 x^6+34 x^5+23 x^4+30 x^3+40 x^2+28 x+32$
• $y^2=9 x^6+35 x^5+18 x^4+23 x^3+26 x^2+9 x+10$
• $y^2=38 x^6+13 x^5+27 x^4+35 x^3+21 x^2+24 x+25$

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 4.0.245312.1.

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.ac_ap	$2$	(not in LMFDB)