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

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

Invariants

Base field:	$\F_{41}$
Dimension:	$2$
L-polynomial:	$1 - 33 x^{2} + 1681 x^{4}$
Frobenius angles:	$\pm0.184081270891$, $\pm0.815918729109$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{115})\)
Galois group:	$C_2^2$
Jacobians:	$42$
Isomorphism classes:	50
Cyclic group of points:	yes

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1649$	$2719201$	$4750234724$	$7997781961225$	$13422659106816929$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$42$	$1616$	$68922$	$2830308$	$115856202$	$4750365206$	$194754273882$	$7984926199108$	$327381934393962$	$13422658903481456$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 42 curves (of which all are hyperelliptic):

• $y^2=38 x^6+16 x^5+10 x^4+5 x^3+26 x^2+33 x+33$
• $y^2=23 x^6+14 x^5+19 x^4+30 x^3+33 x^2+34 x+34$
• $y^2=17 x^6+7 x^5+5 x^4+25 x^3+11 x^2+38 x+18$
• $y^2=20 x^6+x^5+30 x^4+27 x^3+25 x^2+23 x+26$
• $y^2=39 x^6+31 x^5+34 x^4+17 x^3+34 x^2+x+23$
• $y^2=29 x^6+22 x^5+40 x^4+20 x^3+40 x^2+6 x+15$
• $y^2=26 x^6+33 x^5+26 x^4+21 x^3+2 x^2+22 x+31$
• $y^2=33 x^6+34 x^5+33 x^4+3 x^3+12 x^2+9 x+22$
• $y^2=31 x^6+22 x^5+19 x^4+33 x^3+18 x^2+11 x+19$
• $y^2=37 x^6+35 x^5+30 x^4+31 x^3+31 x^2+31 x+15$
• $y^2=17 x^6+5 x^5+16 x^4+22 x^3+22 x^2+22 x+8$
• $y^2=22 x^6+21 x^5+14 x^4+3 x^3+4 x^2+31 x+9$
• $y^2=2 x^6+35 x^5+14 x^4+24 x^3+36 x^2+3 x+15$
• $y^2=11 x^6+27 x^5+30 x^4+19 x^3+9 x^2+5 x+12$
• $y^2=25 x^6+39 x^5+16 x^4+32 x^3+13 x^2+30 x+31$
• $y^2=39 x^6+27 x^5+35 x^4+16 x^3+35 x^2+34 x+12$
• $y^2=29 x^6+39 x^5+5 x^4+14 x^3+5 x^2+40 x+31$
• $y^2=37 x^6+9 x^5+40 x^4+36 x^3+33 x^2+12 x+30$
• $y^2=17 x^6+13 x^5+35 x^4+11 x^3+34 x^2+31 x+16$
• $y^2=27 x^6+8 x^5+35 x^4+5 x^3+23 x^2+29 x+1$
• and

• $y^2=39 x^6+7 x^5+5 x^4+30 x^3+15 x^2+10 x+6$
• $y^2=3 x^6+35 x^5+13 x^4+35 x^3+11 x^2+3 x+33$
• $y^2=18 x^6+5 x^5+37 x^4+5 x^3+25 x^2+18 x+34$
• $y^2=3 x^6+26 x^5+38 x^4+32 x^2+25 x+12$
• $y^2=18 x^6+33 x^5+23 x^4+28 x^2+27 x+31$
• $y^2=34 x^6+12 x^5+10 x^4+14 x^3+12 x^2+12 x+2$
• $y^2=40 x^6+31 x^5+19 x^4+2 x^3+31 x^2+31 x+12$
• $y^2=14 x^6+37 x^5+21 x^4+37 x^3+15 x^2+8 x+21$
• $y^2=3 x^6+11 x^5+8 x^4+22 x^3+19 x^2+23 x+27$
• $y^2=18 x^6+25 x^5+7 x^4+9 x^3+32 x^2+15 x+39$
• $y^2=26 x^6+15 x^5+40 x^4+21 x^3+14 x^2+38 x+23$
• $y^2=33 x^6+8 x^5+35 x^4+3 x^3+2 x^2+23 x+15$
• $y^2=10 x^6+18 x^5+12 x^4+21 x^3+37 x^2+12 x+1$
• $y^2=19 x^6+26 x^5+31 x^4+3 x^3+17 x^2+31 x+6$
• $y^2=6 x^6+23 x^5+2 x^4+19 x^3+29 x^2+8 x+16$
• $y^2=4 x^6+40 x^5+26 x^4+30 x^3+x^2+22 x+37$
• $y^2=24 x^6+35 x^5+33 x^4+16 x^3+6 x^2+9 x+17$
• $y^2=36 x^6+11 x^5+31 x^4+11 x^3+37 x^2+34 x+37$
• $y^2=11 x^6+25 x^5+22 x^4+25 x^3+17 x^2+40 x+17$
• $y^2=10 x^6+21 x^5+4 x^4+21 x^3+17 x^2+16 x+40$
• $y^2=19 x^6+3 x^5+24 x^4+3 x^3+20 x^2+14 x+35$
• $y^2=19 x^6+30 x^5+23 x^4+24 x^3+14 x^2+6 x+18$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{41^{2}}$.

Endomorphism algebra over $\F_{41}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{115})\).

Endomorphism algebra over $\overline{\F}_{41}$

The base change of $A$ to $\F_{41^{2}}$ is 1.1681.abh^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-115}) \)$)$

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.ao_fb	$4$	(not in LMFDB)
2.41.a_bh	$4$	(not in LMFDB)
2.41.o_fb	$4$	(not in LMFDB)