Abelian variety isogeny class 2.67.ay_ks over $\F_{67}$

• Label: 2.67.ay_ks
• Base field: $\F_{67}$
• 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:	$\F_{67}$
Dimension:	$2$
L-polynomial:	$( 1 - 12 x + 67 x^{2} )^{2}$
	$1 - 24 x + 278 x^{2} - 1608 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.238111713333$, $\pm0.238111713333$
Angle rank:	$1$ (numerical)
Jacobians:	$39$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3136$	$20070400$	$90870896704$	$406425600000000$	$1823001737727585856$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$44$	$4470$	$302132$	$20168878$	$1350246524$	$90458649510$	$6060706678532$	$406067600523358$	$27206533801990604$	$1822837802581339350$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 39 curves (of which all are hyperelliptic):

• $y^2=8 x^6+10 x^5+42 x^4+35 x^3+4 x^2+35 x+17$
• $y^2=58 x^6+24 x^5+28 x^4+5 x^3+28 x^2+24 x+58$
• $y^2=16 x^5+38 x^4+40 x^3+61 x^2+21 x$
• $y^2=19 x^5+32 x^4+16 x^3+32 x^2+19 x$
• $y^2=18 x^6+29 x^5+65 x^4+25 x^3+65 x^2+29 x+18$
• $y^2=29 x^6+54 x^5+19 x^4+17 x^3+15 x^2+55 x+29$
• $y^2=66 x^6+21 x^4+21 x^2+66$
• $y^2=40 x^6+42 x^5+50 x^4+47 x^3+7 x^2+31 x+22$
• $y^2=21 x^6+34 x^5+42 x^4+16 x^3+42 x^2+34 x+21$
• $y^2=3 x^6+22 x^5+28 x^4+43 x^3+28 x^2+22 x+3$
• $y^2=55 x^6+41 x^5+44 x^4+32 x^3+44 x^2+41 x+55$
• $y^2=2 x^6+30 x^5+62 x^4+28 x^3+4 x^2+46 x+44$
• $y^2=11 x^6+28 x^4+28 x^2+11$
• $y^2=36 x^6+16 x^5+15 x^4+30 x^3+59 x^2+47 x+26$
• $y^2=42 x^6+66 x^4+66 x^2+42$
• $y^2=x^6+34 x^5+17 x^4+4 x^3+22 x^2+53 x+15$
• $y^2=32 x^6+7 x^5+15 x^4+3 x^3+15 x^2+7 x+32$
• $y^2=30 x^6+57 x^5+66 x^4+61 x^3+7 x^2+46 x+28$
• $y^2=2 x^6+26 x^5+52 x^4+65 x^3+52 x^2+26 x+2$
• $y^2=48 x^6+4 x^5+55 x^4+13 x^3+51 x^2+39 x+1$
• and

• $y^2=12 x^6+62 x^5+34 x^4+49 x^3+34 x^2+62 x+12$
• $y^2=38 x^6+24 x^5+43 x^4+19 x^3+12 x^2+41 x+35$
• $y^2=40 x^6+51 x^5+20 x^4+43 x^3+20 x^2+51 x+40$
• $y^2=10 x^6+45 x^5+46 x^4+64 x^3+53 x^2+5 x+24$
• $y^2=42 x^6+29 x^5+21 x^4+19 x^3+21 x^2+29 x+42$
• $y^2=45 x^6+54 x^5+4 x^4+35 x^3+17 x^2+29 x+5$
• $y^2=45 x^6+26 x^5+53 x^4+38 x^3+53 x^2+26 x+45$
• $y^2=45 x^6+51 x^5+18 x^4+23 x^3+27 x^2+31 x+43$
• $y^2=62 x^6+50 x^5+50 x^4+20 x^3+9 x^2+32 x+39$
• $y^2=11 x^6+51 x^5+63 x^4+36 x^3+44 x^2+7 x+32$
• $y^2=42 x^6+21 x^5+22 x^4+51 x^3+56 x^2+22 x+45$
• $y^2=x^6+6 x^5+11 x^4+26 x^3+11 x^2+6 x+1$
• $y^2=35 x^6+42 x^4+42 x^2+35$
• $y^2=34 x^6+33 x^5+21 x^4+17 x^3+21 x^2+33 x+34$
• $y^2=41 x^6+17 x^5+49 x^4+24 x^3+56 x^2+14 x+11$
• $y^2=66 x^6+13 x^5+37 x^4+8 x^3+24 x^2+11 x+45$
• $y^2=19 x^6+47 x^4+47 x^2+19$
• $y^2=3 x^6+x^4+x^2+3$
• $y^2=54 x^6+15 x^5+46 x^4+66 x^3+46 x^2+15 x+54$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{67}$.

Endomorphism algebra over $\F_{67}$

The isogeny class factors as 1.67.am^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-31}) \)$)$

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
2.67.a_ak	$2$	(not in LMFDB)
2.67.y_ks	$2$	(not in LMFDB)
2.67.m_cz	$3$	(not in LMFDB)