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

• Label: 2.67.o_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:	$\pm0.640638367129$, $\pm0.640638367129$
Angle rank:	$1$ (numerical)
Jacobians:	$44$
Cyclic group of points:	no
Non-cyclic primes:	$3, 5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5625$	$20930625$	$89820090000$	$406138370765625$	$1822997171402015625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$82$	$4660$	$298636$	$20154628$	$1350243142$	$90457321030$	$6060711125026$	$406067752015108$	$27206533907265652$	$1822837802986249300$

Jacobians and polarizations

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

• $y^2=23 x^6+5 x^5+41 x^4+4 x^3+41 x^2+5 x+23$
• $y^2=16 x^6+9 x^5+7 x^4+2 x^3+7 x^2+9 x+16$
• $y^2=21 x^6+34 x^5+13 x^4+22 x^3+13 x^2+34 x+21$
• $y^2=48 x^6+57 x^5+63 x^4+8 x^3+34 x^2+25 x+37$
• $y^2=61 x^6+7 x^5+35 x^4+47 x^3+3 x^2+10 x+61$
• $y^2=51 x^6+33 x^4+41 x^3+20 x^2+58 x+12$
• $y^2=65 x^6+24 x^5+28 x^4+52 x^3+43 x^2+34 x+33$
• $y^2=14 x^6+49 x^5+47 x^4+49 x^3+47 x^2+49 x+14$
• $y^2=21 x^6+42 x^5+47 x^4+27 x^3+47 x^2+42 x+21$
• $y^2=x^6+x^3+1$
• $y^2=52 x^6+42 x^5+57 x^4+53 x^3+9 x^2+63 x+4$
• $y^2=62 x^6+16 x^5+23 x^4+13 x^3+23 x^2+16 x+62$
• $y^2=43 x^6+26 x^5+11 x^4+27 x^3+34 x^2+9 x+47$
• $y^2=65 x^6+30 x^5+49 x^4+56 x^3+49 x^2+30 x+65$
• $y^2=27 x^6+43 x^5+42 x^4+22 x^3+7 x^2+21 x+6$
• $y^2=17 x^6+54 x^5+19 x^4+56 x^3+41 x^2+42 x+13$
• $y^2=37 x^6+42 x^5+62 x^4+40 x^3+62 x^2+42 x+37$
• $y^2=58 x^6+5 x^5+18 x^4+55 x^3+18 x^2+5 x+58$
• $y^2=17 x^6+42 x^5+42 x^4+37 x^3+42 x^2+42 x+17$
• $y^2=22 x^6+9 x^5+55 x^4+7 x^3+x^2+40 x+32$
• and

• $y^2=11 x^6+46 x^5+19 x^4+54 x^3+62 x^2+49 x+14$
• $y^2=13 x^6+65 x^5+24 x^4+12 x^3+24 x^2+65 x+13$
• $y^2=54 x^6+20 x^5+37 x^4+53 x^3+37 x^2+20 x+54$
• $y^2=19 x^6+18 x^5+27 x^4+47 x^3+34 x^2+11 x+3$
• $y^2=x^6+55 x^3+14$
• $y^2=15 x^6+9 x^5+59 x^4+36 x^3+59 x^2+9 x+15$
• $y^2=44 x^6+25 x^5+44 x^4+58 x^3+46 x^2+26 x+34$
• $y^2=31 x^6+30 x^5+39 x^4+59 x^3+24 x^2+61 x+64$
• $y^2=65 x^6+50 x^5+53 x^4+38 x^3+53 x^2+50 x+65$
• $y^2=40 x^6+28 x^5+45 x^4+7 x^3+45 x^2+28 x+40$
• $y^2=57 x^6+2 x^5+15 x^4+49 x^2+15 x+27$
• $y^2=35 x^6+26 x^5+32 x^4+25 x^3+26 x^2+62 x+13$
• $y^2=x^6+45 x^5+32 x^4+16 x^3+6 x^2+49 x+56$
• $y^2=59 x^6+61 x^5+16 x^4+9 x^3+16 x^2+61 x+59$
• $y^2=41 x^6+50 x^5+39 x^4+46 x^3+39 x^2+50 x+41$
• $y^2=27 x^6+17 x^5+44 x^4+61 x^3+2 x^2+35 x+62$
• $y^2=5 x^6+55 x^5+48 x^4+20 x^3+40 x^2+7 x+6$
• $y^2=45 x^6+4 x^5+62 x^4+40 x^3+6 x^2+55 x+64$
• $y^2=35 x^6+28 x^5+10 x^4+54 x^3+10 x^2+28 x+35$
• $y^2=51 x^6+60 x^5+10 x^4+62 x^3+58 x^2+61 x+29$
• $y^2=60 x^6+21 x^5+22 x^4+5 x^3+51 x^2+47 x+22$
• $y^2=49 x^6+64 x^5+43 x^4+x^3+43 x^2+64 x+49$
• $y^2=64 x^6+61 x^5+51 x^4+42 x^3+51 x^2+61 x+64$
• $y^2=24 x^6+x^5+20 x^4+19 x^3+23 x^2+6 x+14$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{67}$

The isogeny class factors as 1.67.h^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.ao_hb	$2$	(not in LMFDB)
2.67.a_dh	$2$	(not in LMFDB)
2.67.ah_as	$3$	(not in LMFDB)