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

• Label: 2.41.ag_dn
• 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:	$( 1 - 3 x + 41 x^{2} )^{2}$
	$1 - 6 x + 91 x^{2} - 246 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.424733087678$, $\pm0.424733087678$
Angle rank:	$1$ (numerical)
Jacobians:	$34$
Cyclic group of points:	no
Non-cyclic primes:	$3, 13$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1521$	$3080025$	$4797501696$	$7973818202025$	$13418043532565841$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$1828$	$69606$	$2821828$	$115816356$	$4750145998$	$194756032836$	$7984928793988$	$327381872971446$	$13422658979725348$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 34 curves (of which all are hyperelliptic):

• $y^2=38 x^6+13 x^5+21 x^4+27 x^3+29 x^2+31 x+34$
• $y^2=26 x^6+16 x^5+11 x^4+36 x^3+11 x^2+16 x+26$
• $y^2=x^6+37 x^5+40 x^4+34 x^3+32 x^2+4 x+32$
• $y^2=29 x^6+3 x^4+24 x^3+22 x^2+19$
• $y^2=2 x^6+6 x^5+5 x^4+31 x^2+24 x+25$
• $y^2=26 x^6+12 x^4+x^3+4 x^2+17 x+31$
• $y^2=39 x^6+29 x^5+13 x^4+33 x^3+14 x^2+30 x+3$
• $y^2=20 x^6+36 x^5+5 x^4+33 x^3+13 x^2+9 x+23$
• $y^2=38 x^6+16 x^5+8 x^4+11 x^3+25 x^2+23 x+24$
• $y^2=36 x^6+39 x^5+34 x^4+30 x^3+34 x^2+39 x+36$
• $y^2=35 x^6+37 x^5+19 x^4+13 x^3+12 x^2+31 x+38$
• $y^2=25 x^6+37 x^5+3 x^4+40 x^3+35 x^2+26 x+36$
• $y^2=32 x^6+25 x^5+x^4+35 x^3+40 x^2+3 x+1$
• $y^2=31 x^6+9 x^5+7 x^4+34 x^3+16 x^2+18 x+25$
• $y^2=21 x^6+33 x^5+27 x^4+12 x^3+17 x^2+20 x+33$
• $y^2=24 x^6+13 x^5+39 x^3+28 x^2+39 x+24$
• $y^2=30 x^6+32 x^5+30 x^4+19 x^3+7 x^2+36 x+13$
• $y^2=11 x^6+16 x^5+13 x^4+23 x^3+13 x^2+7 x+24$
• $y^2=9 x^6+34 x^5+30 x^4+6 x^3+11 x^2+34 x+32$
• $y^2=3 x^6+6 x^5+2 x^4+21 x^3+15 x^2+16 x+9$
• and

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{41}$

The isogeny class factors as 1.41.ad^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-155}) \)$)$

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_cv	$2$	(not in LMFDB)
2.41.g_dn	$2$	(not in LMFDB)
2.41.d_abg	$3$	(not in LMFDB)