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

• Label: 2.67.b_aco
• Base field: $\F_{67}$
• 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:	$\F_{67}$
Dimension:	$2$
L-polynomial:	$1 + x - 66 x^{2} + 67 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.186122649939$, $\pm0.852789316606$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{89})\)
Galois group:	$C_2^2$
Jacobians:	$48$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4492$	$19567152$	$90579329296$	$406243269710784$	$1822867656306795652$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$69$	$4357$	$301164$	$20159833$	$1350147219$	$90459505222$	$6060709562361$	$406067713135921$	$27206534051379348$	$1822837802340407557$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 48 curves (of which all are hyperelliptic):

• $y^2=28 x^6+34 x^5+64 x^4+27 x^3+41 x^2+22 x+51$
• $y^2=26 x^6+63 x^5+11 x^4+36 x^3+65 x^2+2 x+4$
• $y^2=7 x^6+41 x^5+36 x^4+34 x^3+22 x^2+4 x+52$
• $y^2=7 x^6+65 x^5+41 x^4+14 x^3+56 x^2+51 x+57$
• $y^2=41 x^6+6 x^5+57 x^4+55 x^3+20 x^2+64 x+7$
• $y^2=10 x^6+15 x^5+31 x^4+4 x^3+38 x^2+7 x+22$
• $y^2=31 x^6+60 x^5+15 x^4+26 x^3+45 x^2+45 x+62$
• $y^2=24 x^6+6 x^5+43 x^4+20 x^3+10 x^2+19 x+12$
• $y^2=46 x^6+54 x^5+32 x^4+13 x^3+60 x^2+2 x+24$
• $y^2=33 x^6+31 x^5+8 x^4+46 x^3+39 x^2+49 x+13$
• $y^2=2 x^6+2 x^3+27$
• $y^2=59 x^6+27 x^5+41 x^4+9 x^3+15 x^2+57 x+46$
• $y^2=17 x^6+32 x^5+52 x^4+49 x^3+41 x^2+57$
• $y^2=24 x^6+4 x^5+23 x^4+34 x^2+57 x+36$
• $y^2=16 x^6+41 x^5+11 x^4+20 x^3+42 x^2+55 x+62$
• $y^2=22 x^6+33 x^5+10 x^4+8 x^3+30 x^2+57 x+21$
• $y^2=32 x^6+48 x^5+20 x^4+62 x^3+10 x^2+3 x+51$
• $y^2=41 x^6+66 x^4+25 x^3+30 x^2+3 x+29$
• $y^2=11 x^6+28 x^5+17 x^4+61 x^3+23 x^2+48 x+6$
• $y^2=21 x^6+52 x^5+12 x^4+x^3+14 x^2+14 x+18$
• and

• $y^2=20 x^6+14 x^5+43 x^4+x^3+23 x^2+38 x+26$
• $y^2=66 x^6+51 x^5+44 x^4+57 x^3+65 x^2+46 x+14$
• $y^2=12 x^6+50 x^5+25 x^4+7 x^3+29 x^2+41 x+18$
• $y^2=14 x^6+30 x^5+27 x^4+55 x^3+48 x^2+58 x+54$
• $y^2=6 x^6+x^5+6 x^4+37 x^3+34 x^2+23 x+42$
• $y^2=39 x^6+43 x^5+56 x^4+9 x^3+20 x^2+37 x+41$
• $y^2=47 x^6+47 x^5+52 x^4+16 x^3+63 x^2+33 x+35$
• $y^2=21 x^6+30 x^5+42 x^4+58 x^3+47 x^2+63 x+29$
• $y^2=45 x^6+61 x^5+50 x^4+61 x^3+63 x^2+20 x+7$
• $y^2=25 x^6+21 x^5+5 x^4+27 x^3+20 x^2+7 x+12$
• $y^2=48 x^6+57 x^5+3 x^4+41 x^3+12 x^2+24 x+11$
• $y^2=30 x^6+65 x^5+22 x^4+x^3+41 x^2+24 x+47$
• $y^2=65 x^6+2 x^5+39 x^4+3 x^3+27 x^2+63 x+14$
• $y^2=9 x^6+45 x^5+13 x^4+52 x^3+42 x^2+14 x+22$
• $y^2=43 x^6+41 x^5+65 x^4+49 x^3+25 x^2+19 x+55$
• $y^2=27 x^6+10 x^5+20 x^4+13 x^3+40 x^2+42 x+40$
• $y^2=42 x^6+8 x^5+11 x^4+36 x^3+54 x^2+65 x+33$
• $y^2=31 x^6+42 x^4+60 x^3+51 x^2+41 x+55$
• $y^2=64 x^6+43 x^5+12 x^4+28 x^3+2 x^2+65 x+34$
• $y^2=56 x^6+28 x^5+46 x^4+46 x^3+21 x^2+64 x+20$
• $y^2=44 x^6+54 x^5+12 x^4+26 x^3+19 x^2+32 x+58$
• $y^2=25 x^6+43 x^5+40 x^4+37 x^3+58 x^2+3 x+43$
• $y^2=29 x^6+54 x^5+33 x^4+41 x^3+12 x^2+2 x+25$
• $y^2=33 x^6+14 x^5+9 x^4+x^3+11 x^2+22 x+41$
• $y^2=28 x^6+54 x^5+15 x^4+x^3+7 x^2+6 x+33$
• $y^2=11 x^6+59 x^5+61 x^4+18 x^3+2 x^2+13 x+23$
• $y^2=11 x^6+39 x^5+36 x^4+48 x^3+5 x^2+39 x+40$
• $y^2=44 x^6+55 x^5+37 x^4+10 x^3+3 x^2+12 x+18$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{67^{3}}$.

Endomorphism algebra over $\F_{67}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{89})\).

Endomorphism algebra over $\overline{\F}_{67}$

The base change of $A$ to $\F_{67^{3}}$ is 1.300763.hs^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-267}) \)$)$

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.ab_aco	$2$	(not in LMFDB)
2.67.ac_ff	$3$	(not in LMFDB)
2.67.ab_aco	$6$	(not in LMFDB)