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

• Label: 2.41.as_gg
• 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:	$( 1 - 10 x + 41 x^{2} )( 1 - 8 x + 41 x^{2} )$
	$1 - 18 x + 162 x^{2} - 738 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.214776712523$, $\pm0.285223287477$
Angle rank:	$1$ (numerical)
Jacobians:	$10$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1088$	$2828800$	$4798733888$	$8002109440000$	$13425660552953408$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$1682$	$69624$	$2831838$	$115882104$	$4750104242$	$194753381784$	$7984918073278$	$327381906567384$	$13422659310152402$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{41^{4}}$.

Endomorphism algebra over $\F_{41}$

The isogeny class factors as 1.41.ak $\times$ 1.41.ai and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.41.ak : \(\Q(\sqrt{-1}) \).
• 1.41.ai : \(\Q(\sqrt{-1}) \).

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

The base change of $A$ to $\F_{41^{4}}$ is 1.2825761.emw^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{41^{2}}$
  The base change of $A$ to $\F_{41^{2}}$ is 1.1681.as $\times$ 1.1681.s. The endomorphism algebra for each factor is:

  • 1.1681.as : \(\Q(\sqrt{-1}) \).
  • 1.1681.s : \(\Q(\sqrt{-1}) \).

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.ac_c	$2$	(not in LMFDB)
2.41.c_c	$2$	(not in LMFDB)
2.41.s_gg	$2$	(not in LMFDB)