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

• Label: 2.41.ad_cu
• 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:	$( 1 - 5 x + 41 x^{2} )( 1 + 2 x + 41 x^{2} )$
	$1 - 3 x + 72 x^{2} - 123 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.372324822061$, $\pm0.549915982954$
Angle rank:	$2$ (numerical)
Jacobians:	$54$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1628$	$3060640$	$4767493808$	$7977558160000$	$13422158398684508$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$39$	$1817$	$69174$	$2823153$	$115851879$	$4750083182$	$194753777439$	$7984929110113$	$327381986457414$	$13422659161250777$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 54 curves (of which all are hyperelliptic):

• $y^2=5 x^6+11 x^5+8 x^4+25 x^3+34 x^2+39 x+33$
• $y^2=4 x^6+36 x^5+12 x^4+8 x^3+15 x^2+25 x+39$
• $y^2=16 x^6+35 x^5+34 x^4+23 x^3+10 x^2+22 x+34$
• $y^2=24 x^6+20 x^5+5 x^4+8 x^3+6 x^2+15 x+10$
• $y^2=12 x^6+7 x^5+15 x^4+4 x^2+13$
• $y^2=37 x^6+28 x^5+16 x^4+13 x^3+x^2+30 x+7$
• $y^2=38 x^6+17 x^5+10 x^4+9 x^3+15 x^2+27 x+24$
• $y^2=3 x^6+33 x^5+29 x^4+22 x^3+2 x^2+14 x+16$
• $y^2=17 x^6+12 x^5+30 x^4+38 x^3+25 x^2+2 x+38$
• $y^2=6 x^6+15 x^5+14 x^4+22 x^3+28 x^2+17 x+1$
• $y^2=15 x^6+30 x^5+36 x^4+15 x^3+5 x^2+40 x+10$
• $y^2=15 x^6+15 x^5+30 x^4+16 x^3+40 x^2+10 x+8$
• $y^2=8 x^6+3 x^5+25 x^4+21 x^3+39 x^2+8 x+34$
• $y^2=29 x^6+8 x^5+39 x^4+26 x^3+19 x^2+4 x+27$
• $y^2=7 x^6+33 x^5+5 x^4+23 x^3+35 x^2+35 x$
• $y^2=30 x^6+6 x^5+20 x^4+39 x^3+23 x^2+2 x+23$
• $y^2=26 x^6+40 x^5+6 x^4+29 x^3+14 x^2+16 x+31$
• $y^2=34 x^6+7 x^5+14 x^4+13 x^3+16 x^2+10 x+31$
• $y^2=34 x^6+29 x^5+37 x^4+27 x^3+24 x^2+6 x+8$
• $y^2=19 x^6+15 x^5+2 x^4+40 x^2+13 x+35$
• and

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

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 $\times$ 1.41.c 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.af : \(\Q(\sqrt{-139}) \).
• 1.41.c : \(\Q(\sqrt{-10}) \).

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.ah_do	$2$	(not in LMFDB)
2.41.d_cu	$2$	(not in LMFDB)
2.41.h_do	$2$	(not in LMFDB)