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

• Label: 2.67.d_acg
• 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:	$1 + 3 x - 58 x^{2} + 201 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.225329797192$, $\pm0.891996463858$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-259})\)
Galois group:	$C_2^2$
Jacobians:	$44$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4636$	$19601008$	$90805795600$	$406201646096064$	$1822916833500694996$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$71$	$4365$	$301916$	$20157769$	$1350183641$	$90458921670$	$6060706874363$	$406067681437009$	$27206533739063972$	$1822837805277623325$

Jacobians and polarizations

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

• $y^2=5 x^5+48 x^4+52 x^3+58 x^2+10 x+25$
• $y^2=12 x^6+38 x^5+46 x^4+61 x^3+53 x^2+58 x+47$
• $y^2=59 x^6+27 x^5+13 x^4+55 x^3+28 x^2+30 x$
• $y^2=9 x^6+15 x^5+62 x^4+51 x^3+9 x^2+28 x+8$
• $y^2=24 x^6+40 x^5+27 x^4+36 x^3+10 x^2+50 x+50$
• $y^2=65 x^5+58 x^4+47 x^3+8 x^2+32 x+61$
• $y^2=27 x^6+52 x^5+57 x^4+42 x^3+33 x^2+49 x+62$
• $y^2=20 x^6+25 x^4+20 x^3+50 x^2+9 x+45$
• $y^2=7 x^5+17 x^4+x^3+7 x^2+x+66$
• $y^2=34 x^6+x^5+20 x^4+3 x^3+19 x^2+63 x+61$
• $y^2=50 x^6+18 x^5+29 x^4+37 x^3+3 x^2+52 x+44$
• $y^2=63 x^6+59 x^5+13 x^4+20 x^3+32 x^2+62 x+1$
• $y^2=4 x^6+24 x^5+56 x^3+5 x^2+45 x$
• $y^2=44 x^6+44 x^5+28 x^4+61 x^3+31 x^2+27 x+57$
• $y^2=63 x^6+24 x^5+19 x^4+20 x^3+51 x^2+42 x+41$
• $y^2=60 x^6+61 x^5+18 x^4+40 x^3+29 x^2+24 x$
• $y^2=63 x^6+4 x^5+56 x^4+25 x^2+41 x+18$
• $y^2=61 x^6+60 x^5+48 x^4+30 x^3+55 x^2+47 x+42$
• $y^2=31 x^6+42 x^5+51 x^4+46 x^3+4 x^2+29 x+27$
• $y^2=32 x^6+15 x^5+55 x^4+41 x^3+13 x^2+11 x+47$
• and

• $y^2=62 x^6+11 x^5+22 x^4+33 x^3+33 x^2+47 x+58$
• $y^2=34 x^6+15 x^5+32 x^4+47 x^3+48 x^2+26 x+8$
• $y^2=22 x^6+7 x^5+47 x^4+45 x^3+62 x^2+17 x+15$
• $y^2=22 x^6+48 x^5+10 x^4+30 x^3+63 x^2+32 x+30$
• $y^2=2 x^6+59 x^5+25 x^4+9 x^3+56 x^2+3 x+26$
• $y^2=16 x^6+30 x^5+49 x^4+15 x^3+32 x^2+4 x+12$
• $y^2=16 x^5+27 x^4+5 x^3+48 x^2+49 x+62$
• $y^2=43 x^6+61 x^5+50 x^4+43 x^3+35 x^2+64 x+56$
• $y^2=44 x^6+59 x^5+28 x^4+10 x^3+x^2+6 x+9$
• $y^2=51 x^6+10 x^5+28 x^4+64 x^3+65 x^2+29 x+35$
• $y^2=49 x^6+39 x^5+34 x^4+43 x^3+60 x^2+55 x+9$
• $y^2=27 x^6+54 x^5+52 x^4+61 x^3+30 x^2+6 x+12$
• $y^2=63 x^6+39 x^5+16 x^4+52 x^3+2 x^2+44 x+36$
• $y^2=18 x^6+46 x^5+32 x^4+39 x^2+64 x+62$
• $y^2=7 x^6+6 x^5+62 x^4+22 x^3+62 x^2+58 x+55$
• $y^2=51 x^6+29 x^5+8 x^4+45 x^3+15 x+37$
• $y^2=50 x^6+55 x^5+18 x^4+15 x^3+53 x^2+47 x+63$
• $y^2=53 x^6+48 x^5+21 x^4+53 x^3+32 x^2+63 x+34$
• $y^2=66 x^6+59 x^5+30 x^4+47 x^3+2 x^2+21 x+37$
• $y^2=57 x^6+59 x^5+9 x^4+52 x^3+48 x^2+15 x+53$
• $y^2=26 x^6+28 x^5+29 x^4+14 x^3+19 x^2+37 x+27$
• $y^2=53 x^6+60 x^5+65 x^4+55 x^3+7 x^2+33 x+22$
• $y^2=15 x^6+39 x^5+61 x^4+34 x^3+23 x^2+29 x+51$
• $y^2=33 x^6+63 x^5+29 x^4+12 x^3+41 x^2+57 x+17$

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 \(\Q(\sqrt{-3}, \sqrt{-259})\).

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

The base change of $A$ to $\F_{67^{3}}$ is 1.300763.we^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-259}) \)$)$

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.ad_acg	$2$	(not in LMFDB)
2.67.ag_fn	$3$	(not in LMFDB)
2.67.a_ev	$6$	(not in LMFDB)