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

• Label: 2.41.ak_ed
• Base field: $\F_{41}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{41}$
Dimension:	$2$
L-polynomial:	$( 1 - 5 x + 41 x^{2} )^{2}$
	$1 - 10 x + 107 x^{2} - 410 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.372324822061$, $\pm0.372324822061$
Angle rank:	$1$ (numerical)
Jacobians:	$21$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1369$	$3024121$	$4818025744$	$7985569515625$	$13418135738402329$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$32$	$1796$	$69902$	$2825988$	$115817152$	$4749899726$	$194754852352$	$7984936506628$	$327381967064222$	$13422659011125956$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 21 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{41}$.

Endomorphism algebra over $\F_{41}$

The isogeny class factors as 1.41.af^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-139}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.41.a_cf	$2$	(not in LMFDB)
2.41.k_ed	$2$	(not in LMFDB)
2.41.f_aq	$3$	(not in LMFDB)