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

• Label: 2.67.ab_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:	$\pm0.147210683394$, $\pm0.813877350061$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{89})\)
Galois group:	$C_2^2$
Jacobians:	$48$
Isomorphism classes:	88
Cyclic group of points:	no
Non-cyclic primes:	$2, 3, 11$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4356$	$19567152$	$90338718096$	$406243269710784$	$1822807951074269676$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$67$	$4357$	$300364$	$20159833$	$1350102997$	$90459505222$	$6060713648287$	$406067713135921$	$27206534741210548$	$1822837802340407557$

Jacobians and polarizations

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

• $y^2=56 x^6+x^5+61 x^4+54 x^3+15 x^2+44 x+35$
• $y^2=13 x^6+65 x^5+39 x^4+18 x^3+66 x^2+x+2$
• $y^2=37 x^6+54 x^5+18 x^4+17 x^3+11 x^2+2 x+26$
• $y^2=37 x^6+66 x^5+54 x^4+7 x^3+28 x^2+59 x+62$
• $y^2=54 x^6+3 x^5+62 x^4+61 x^3+10 x^2+32 x+37$
• $y^2=5 x^6+41 x^5+49 x^4+2 x^3+19 x^2+37 x+11$
• $y^2=62 x^6+53 x^5+30 x^4+52 x^3+23 x^2+23 x+57$
• $y^2=12 x^6+3 x^5+55 x^4+10 x^3+5 x^2+43 x+6$
• $y^2=23 x^6+27 x^5+16 x^4+40 x^3+30 x^2+x+12$
• $y^2=50 x^6+49 x^5+4 x^4+23 x^3+53 x^2+58 x+40$
• $y^2=x^6+x^3+47$
• $y^2=51 x^6+54 x^5+15 x^4+18 x^3+30 x^2+47 x+25$
• $y^2=42 x^6+16 x^5+26 x^4+58 x^3+54 x^2+62$
• $y^2=48 x^6+8 x^5+46 x^4+x^2+47 x+5$
• $y^2=8 x^6+54 x^5+39 x^4+10 x^3+21 x^2+61 x+31$
• $y^2=44 x^6+66 x^5+20 x^4+16 x^3+60 x^2+47 x+42$
• $y^2=64 x^6+29 x^5+40 x^4+57 x^3+20 x^2+6 x+35$
• $y^2=54 x^6+33 x^4+46 x^3+15 x^2+35 x+48$
• $y^2=22 x^6+56 x^5+34 x^4+55 x^3+46 x^2+29 x+12$
• $y^2=42 x^6+37 x^5+24 x^4+2 x^3+28 x^2+28 x+36$
• and

• $y^2=40 x^6+28 x^5+19 x^4+2 x^3+46 x^2+9 x+52$
• $y^2=33 x^6+59 x^5+22 x^4+62 x^3+66 x^2+23 x+7$
• $y^2=6 x^6+25 x^5+46 x^4+37 x^3+48 x^2+54 x+9$
• $y^2=28 x^6+60 x^5+54 x^4+43 x^3+29 x^2+49 x+41$
• $y^2=3 x^6+34 x^5+3 x^4+52 x^3+17 x^2+45 x+21$
• $y^2=53 x^6+55 x^5+28 x^4+38 x^3+10 x^2+52 x+54$
• $y^2=27 x^6+27 x^5+37 x^4+32 x^3+59 x^2+66 x+3$
• $y^2=42 x^6+60 x^5+17 x^4+49 x^3+27 x^2+59 x+58$
• $y^2=23 x^6+55 x^5+33 x^4+55 x^3+59 x^2+40 x+14$
• $y^2=50 x^6+42 x^5+10 x^4+54 x^3+40 x^2+14 x+24$
• $y^2=29 x^6+47 x^5+6 x^4+15 x^3+24 x^2+48 x+22$
• $y^2=60 x^6+63 x^5+44 x^4+2 x^3+15 x^2+48 x+27$
• $y^2=66 x^6+x^5+53 x^4+35 x^3+47 x^2+65 x+7$
• $y^2=38 x^6+56 x^5+40 x^4+26 x^3+21 x^2+7 x+11$
• $y^2=55 x^6+54 x^5+66 x^4+58 x^3+46 x^2+43 x+61$
• $y^2=47 x^6+5 x^5+10 x^4+40 x^3+20 x^2+21 x+20$
• $y^2=17 x^6+16 x^5+22 x^4+5 x^3+41 x^2+63 x+66$
• $y^2=62 x^6+17 x^4+53 x^3+35 x^2+15 x+43$
• $y^2=61 x^6+19 x^5+24 x^4+56 x^3+4 x^2+63 x+1$
• $y^2=45 x^6+56 x^5+25 x^4+25 x^3+42 x^2+61 x+40$
• $y^2=21 x^6+41 x^5+24 x^4+52 x^3+38 x^2+64 x+49$
• $y^2=50 x^6+19 x^5+13 x^4+7 x^3+49 x^2+6 x+19$
• $y^2=58 x^6+41 x^5+66 x^4+15 x^3+24 x^2+4 x+50$
• $y^2=66 x^6+28 x^5+18 x^4+2 x^3+22 x^2+44 x+15$
• $y^2=14 x^6+27 x^5+41 x^4+34 x^3+37 x^2+3 x+50$
• $y^2=39 x^6+63 x^5+64 x^4+9 x^3+x^2+40 x+45$
• $y^2=39 x^6+53 x^5+18 x^4+24 x^3+36 x^2+53 x+20$
• $y^2=21 x^6+43 x^5+7 x^4+20 x^3+6 x^2+24 x+36$

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.ahs^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.b_aco	$2$	(not in LMFDB)
2.67.c_ff	$3$	(not in LMFDB)
2.67.ac_ff	$6$	(not in LMFDB)