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

• Label: 2.41.a_acs
• 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:	$1 - 70 x^{2} + 1681 x^{4}$
Frobenius angles:	$\pm0.0871893050776$, $\pm0.912810694922$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{38})\)
Galois group:	$C_2^2$
Jacobians:	$20$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1612$	$2598544$	$4750114252$	$7976241202176$	$13422659523351052$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$42$	$1542$	$68922$	$2822686$	$115856202$	$4750124262$	$194754273882$	$7984931801278$	$327381934393962$	$13422659736549702$

Jacobians and polarizations

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

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

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 \(\Q(\sqrt{-3}, \sqrt{38})\).

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

The base change of $A$ to $\F_{41^{2}}$ is 1.1681.acs^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-114}) \)$)$

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.ag_cb	$3$	(not in LMFDB)
2.41.g_cb	$3$	(not in LMFDB)
2.41.a_cs	$4$	(not in LMFDB)