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

• Label: 2.41.d_abg
• 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:	$1 + 3 x - 32 x^{2} + 123 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.241933578988$, $\pm0.908600245655$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-155})\)
Galois group:	$C_2^2$
Jacobians:	$44$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1776$	$2706624$	$4797501696$	$7990484546304$	$13424967794334576$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$45$	$1609$	$69606$	$2827729$	$115876125$	$4750145998$	$194753394405$	$7984923446689$	$327381872971446$	$13422659475365929$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 44 curves (of which all are hyperelliptic):

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

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

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 \(\Q(\sqrt{-3}, \sqrt{-155})\).

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

The base change of $A$ to $\F_{41^{3}}$ is 1.68921.ne^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-155}) \)$)$

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.ad_abg	$2$	(not in LMFDB)
2.41.ag_dn	$3$	(not in LMFDB)
2.41.a_cv	$6$	(not in LMFDB)