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

• Label: 2.41.au_gw
• 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:	$( 1 - 12 x + 41 x^{2} )( 1 - 8 x + 41 x^{2} )$
	$1 - 20 x + 178 x^{2} - 820 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.113551764296$, $\pm0.285223287477$
Angle rank:	$2$ (numerical)
Jacobians:	$20$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1020$	$2754000$	$4765285980$	$7992152064000$	$13423760198305500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$1638$	$69142$	$2828318$	$115865702$	$4750093638$	$194754096902$	$7984927070398$	$327381978057142$	$13422659728370598$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{41}$

The isogeny class factors as 1.41.am $\times$ 1.41.ai and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.41.am : \(\Q(\sqrt{-5}) \).
• 1.41.ai : \(\Q(\sqrt{-1}) \).

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.ae_ao	$2$	(not in LMFDB)
2.41.e_ao	$2$	(not in LMFDB)
2.41.u_gw	$2$	(not in LMFDB)