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

• Label: 2.67.k_gd
• 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:	$( 1 + 5 x + 67 x^{2} )^{2}$
	$1 + 10 x + 159 x^{2} + 670 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.598798062002$, $\pm0.598798062002$
Angle rank:	$1$ (numerical)
Jacobians:	$43$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5329$	$21150801$	$89930413456$	$405950728871961$	$1823036213535207889$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$78$	$4708$	$299004$	$20145316$	$1350272058$	$90458036422$	$6060703488414$	$406067741306308$	$27206534621379588$	$1822837799155110628$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 43 curves (of which all are hyperelliptic):

• $y^2=23 x^6+29 x^5+24 x^4+5 x^3+24 x^2+29 x+23$
• $y^2=57 x^6+65 x^5+34 x^4+41 x^3+50 x^2+28 x+24$
• $y^2=46 x^6+20 x^5+24 x^4+6 x^3+20 x^2+51 x+49$
• $y^2=24 x^6+17 x^4+65 x^3+17 x^2+24$
• $y^2=15 x^6+10 x^4+11 x^3+10 x^2+15$
• $y^2=25 x^6+12 x^5+49 x^4+59 x^3+49 x^2+12 x+25$
• $y^2=15 x^6+39 x^5+52 x^4+65 x^3+52 x^2+39 x+15$
• $y^2=26 x^6+28 x^5+63 x^4+30 x^3+63 x^2+28 x+26$
• $y^2=10 x^6+15 x^5+57 x^4+29 x^3+57 x^2+15 x+10$
• $y^2=44 x^6+6 x^5+35 x^4+62 x^3+48 x^2+4 x+54$
• $y^2=40 x^6+38 x^5+11 x^4+7 x^3+11 x^2+38 x+40$
• $y^2=48 x^6+40 x^5+19 x^4+37 x^3+19 x^2+40 x+48$
• $y^2=41 x^6+4 x^5+25 x^4+18 x^3+45 x^2+33 x+56$
• $y^2=56 x^6+31 x^5+27 x^4+32 x^3+x^2+62 x+60$
• $y^2=35 x^6+56 x^5+23 x^4+17 x^3+23 x^2+56 x+35$
• $y^2=59 x^6+27 x^5+61 x^4+26 x^3+61 x^2+27 x+59$
• $y^2=37 x^6+11 x^5+61 x^4+66 x^3+61 x^2+11 x+37$
• $y^2=25 x^6+24 x^5+18 x^4+54 x^3+17 x^2+3 x+45$
• $y^2=45 x^6+52 x^5+13 x^4+59 x^3+64 x^2+45 x+47$
• $y^2=12 x^6+31 x^5+63 x^4+60 x^3+63 x^2+31 x+12$
• and

• $y^2=38 x^6+59 x^5+60 x^4+44 x^3+60 x^2+59 x+38$
• $y^2=56 x^6+13 x^5+58 x^4+15 x^3+58 x^2+13 x+56$
• $y^2=4 x^6+64 x^5+39 x^4+49 x^3+39 x^2+64 x+4$
• $y^2=39 x^6+58 x^5+51 x^4+60 x^3+66 x^2+6 x+8$
• $y^2=2 x^6+2 x^3+50$
• $y^2=2 x^6+3 x^3+57$
• $y^2=64 x^6+3 x^5+50 x^4+22 x^3+17 x^2+41 x+34$
• $y^2=35 x^6+32 x^5+40 x^4+33 x^3+66 x^2+62 x+54$
• $y^2=49 x^6+35 x^5+12 x^4+32 x^3+12 x^2+35 x+49$
• $y^2=26 x^6+57 x^5+x^4+43 x^3+x^2+57 x+26$
• $y^2=65 x^6+14 x^5+52 x^4+9 x^3+16 x^2+11 x+10$
• $y^2=7 x^6+43 x^5+8 x^4+20 x^3+23 x^2+63 x+18$
• $y^2=17 x^6+64 x^5+26 x^4+47 x^3+16 x^2+55 x+7$
• $y^2=57 x^6+40 x^5+15 x^4+59 x^3+15 x^2+40 x+57$
• $y^2=8 x^6+22 x^5+43 x^2+52 x+6$
• $y^2=2 x^6+63 x^3+51$
• $y^2=2 x^6+38 x^3+50$
• $y^2=3 x^6+20 x^5+61 x^4+43 x^3+49 x^2+5 x+15$
• $y^2=53 x^6+26 x^5+22 x^4+7 x^3+62 x^2+51 x+6$
• $y^2=22 x^6+24 x^5+41 x^4+55 x^3+41 x^2+24 x+22$
• $y^2=61 x^6+29 x^5+6 x^4+24 x^3+6 x^2+29 x+61$
• $y^2=53 x^6+30 x^5+42 x^4+39 x^3+13 x^2+17 x+10$
• $y^2=43 x^6+29 x^5+9 x^4+6 x^3+9 x^2+29 x+43$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{67}$

The isogeny class factors as 1.67.f^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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.ak_gd	$2$	(not in LMFDB)
2.67.a_ef	$2$	(not in LMFDB)
2.67.abg_pa	$3$	(not in LMFDB)
2.67.al_cc	$3$	(not in LMFDB)
2.67.af_abq	$3$	(not in LMFDB)
2.67.q_hh	$3$	(not in LMFDB)
2.67.w_jv	$3$	(not in LMFDB)