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

• Label: 2.67.ao_hb
• 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:	$( 1 - 7 x + 67 x^{2} )^{2}$
	$1 - 14 x + 183 x^{2} - 938 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.359361632871$, $\pm0.359361632871$
Angle rank:	$1$ (numerical)
Jacobians:	$44$
Cyclic group of points:	no
Non-cyclic primes:	$61$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3721$	$20930625$	$91100141584$	$406138370765625$	$1822678450068020281$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$54$	$4660$	$302892$	$20154628$	$1350007074$	$90457321030$	$6060712085622$	$406067752015108$	$27206534885324244$	$1822837802986249300$

Jacobians and polarizations

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

• $y^2=45 x^6+36 x^5+54 x^4+2 x^3+54 x^2+36 x+45$
• $y^2=8 x^6+38 x^5+37 x^4+x^3+37 x^2+38 x+8$
• $y^2=42 x^6+x^5+26 x^4+44 x^3+26 x^2+x+42$
• $y^2=24 x^6+62 x^5+65 x^4+4 x^3+17 x^2+46 x+52$
• $y^2=55 x^6+14 x^5+3 x^4+27 x^3+6 x^2+20 x+55$
• $y^2=59 x^6+50 x^4+54 x^3+10 x^2+29 x+6$
• $y^2=66 x^6+12 x^5+14 x^4+26 x^3+55 x^2+17 x+50$
• $y^2=28 x^6+31 x^5+27 x^4+31 x^3+27 x^2+31 x+28$
• $y^2=44 x^6+21 x^5+57 x^4+47 x^3+57 x^2+21 x+44$
• $y^2=2 x^6+2 x^3+2$
• $y^2=37 x^6+17 x^5+47 x^4+39 x^3+18 x^2+59 x+8$
• $y^2=57 x^6+32 x^5+46 x^4+26 x^3+46 x^2+32 x+57$
• $y^2=55 x^6+13 x^5+39 x^4+47 x^3+17 x^2+38 x+57$
• $y^2=63 x^6+60 x^5+31 x^4+45 x^3+31 x^2+60 x+63$
• $y^2=54 x^6+19 x^5+17 x^4+44 x^3+14 x^2+42 x+12$
• $y^2=42 x^6+27 x^5+43 x^4+28 x^3+54 x^2+21 x+40$
• $y^2=7 x^6+17 x^5+57 x^4+13 x^3+57 x^2+17 x+7$
• $y^2=49 x^6+10 x^5+36 x^4+43 x^3+36 x^2+10 x+49$
• $y^2=42 x^6+21 x^5+21 x^4+52 x^3+21 x^2+21 x+42$
• $y^2=44 x^6+18 x^5+43 x^4+14 x^3+2 x^2+13 x+64$
• and

• $y^2=22 x^6+25 x^5+38 x^4+41 x^3+57 x^2+31 x+28$
• $y^2=26 x^6+63 x^5+48 x^4+24 x^3+48 x^2+63 x+26$
• $y^2=27 x^6+10 x^5+52 x^4+60 x^3+52 x^2+10 x+27$
• $y^2=43 x^6+9 x^5+47 x^4+57 x^3+17 x^2+39 x+35$
• $y^2=2 x^6+43 x^3+28$
• $y^2=30 x^6+18 x^5+51 x^4+5 x^3+51 x^2+18 x+30$
• $y^2=22 x^6+46 x^5+22 x^4+29 x^3+23 x^2+13 x+17$
• $y^2=49 x^6+15 x^5+53 x^4+63 x^3+12 x^2+64 x+32$
• $y^2=63 x^6+33 x^5+39 x^4+9 x^3+39 x^2+33 x+63$
• $y^2=20 x^6+14 x^5+56 x^4+37 x^3+56 x^2+14 x+20$
• $y^2=62 x^6+x^5+41 x^4+58 x^2+41 x+47$
• $y^2=3 x^6+52 x^5+64 x^4+50 x^3+52 x^2+57 x+26$
• $y^2=2 x^6+23 x^5+64 x^4+32 x^3+12 x^2+31 x+45$
• $y^2=63 x^6+64 x^5+8 x^4+38 x^3+8 x^2+64 x+63$
• $y^2=54 x^6+25 x^5+53 x^4+23 x^3+53 x^2+25 x+54$
• $y^2=47 x^6+42 x^5+22 x^4+64 x^3+x^2+51 x+31$
• $y^2=36 x^6+61 x^5+24 x^4+10 x^3+20 x^2+37 x+3$
• $y^2=23 x^6+8 x^5+57 x^4+13 x^3+12 x^2+43 x+61$
• $y^2=51 x^6+14 x^5+5 x^4+27 x^3+5 x^2+14 x+51$
• $y^2=35 x^6+53 x^5+20 x^4+57 x^3+49 x^2+55 x+58$
• $y^2=30 x^6+44 x^5+11 x^4+36 x^3+59 x^2+57 x+11$
• $y^2=58 x^6+32 x^5+55 x^4+34 x^3+55 x^2+32 x+58$
• $y^2=61 x^6+55 x^5+35 x^4+17 x^3+35 x^2+55 x+61$
• $y^2=48 x^6+2 x^5+40 x^4+38 x^3+46 x^2+12 x+28$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{67}$

The isogeny class factors as 1.67.ah^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-219}) \)$)$

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_dh	$2$	(not in LMFDB)
2.67.o_hb	$2$	(not in LMFDB)
2.67.h_as	$3$	(not in LMFDB)