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

• Label: 2.41.ai_x
• 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:	$1 - 8 x + 23 x^{2} - 328 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.0481100458561$, $\pm0.618556620811$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\zeta_{12})\)
Galois group:	$C_2^2$
Jacobians:	$20$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1369$	$2794129$	$4685402500$	$7976346967849$	$13423232938730809$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$34$	$1664$	$67978$	$2822724$	$115861154$	$4749934358$	$194753391314$	$7984928807044$	$327381919270138$	$13422659102962304$

Jacobians and polarizations

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

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

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 \(\Q(\zeta_{12})\).

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

The base change of $A$ to $\F_{41^{3}}$ is 1.68921.ase^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.i_x	$2$	(not in LMFDB)
2.41.q_fq	$3$	(not in LMFDB)
2.41.ak_ch	$4$	(not in LMFDB)