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

• Label: 2.41.f_aq
• 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:	$1 + 5 x - 16 x^{2} + 205 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.294341844606$, $\pm0.961008511272$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-139})\)
Galois group:	$C_2^2$
Jacobians:	$15$
Isomorphism classes:	27
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1876$	$2731456$	$4818025744$	$7984603105024$	$13424921667865876$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$47$	$1625$	$69902$	$2825649$	$115875727$	$4749899726$	$194753984647$	$7984919590369$	$327381967064222$	$13422659459665625$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 15 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{41^{3}}$.

Endomorphism algebra over $\F_{41}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-139})\).

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

The base change of $A$ to $\F_{41^{3}}$ is 1.68921.sw^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-139}) \)$)$

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.af_aq	$2$	(not in LMFDB)
2.41.ak_ed	$3$	(not in LMFDB)
2.41.a_cf	$6$	(not in LMFDB)