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

• Label: 2.67.ak_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:	$\pm0.401201937998$, $\pm0.401201937998$
Angle rank:	$1$ (numerical)
Jacobians:	$43$
Cyclic group of points:	no
Non-cyclic primes:	$3, 7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3969$	$21150801$	$90989102736$	$405950728871961$	$1822639411765966689$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$58$	$4708$	$302524$	$20145316$	$1349978158$	$90458036422$	$6060719722234$	$406067741306308$	$27206534171210308$	$1822837799155110628$

Jacobians and polarizations

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

• $y^2=46 x^6+58 x^5+48 x^4+10 x^3+48 x^2+58 x+46$
• $y^2=47 x^6+63 x^5+x^4+15 x^3+33 x^2+56 x+48$
• $y^2=25 x^6+40 x^5+48 x^4+12 x^3+40 x^2+35 x+31$
• $y^2=12 x^6+42 x^4+66 x^3+42 x^2+12$
• $y^2=41 x^6+5 x^4+39 x^3+5 x^2+41$
• $y^2=46 x^6+6 x^5+58 x^4+63 x^3+58 x^2+6 x+46$
• $y^2=30 x^6+11 x^5+37 x^4+63 x^3+37 x^2+11 x+30$
• $y^2=13 x^6+14 x^5+65 x^4+15 x^3+65 x^2+14 x+13$
• $y^2=20 x^6+30 x^5+47 x^4+58 x^3+47 x^2+30 x+20$
• $y^2=21 x^6+12 x^5+3 x^4+57 x^3+29 x^2+8 x+41$
• $y^2=20 x^6+19 x^5+39 x^4+37 x^3+39 x^2+19 x+20$
• $y^2=24 x^6+20 x^5+43 x^4+52 x^3+43 x^2+20 x+24$
• $y^2=15 x^6+8 x^5+50 x^4+36 x^3+23 x^2+66 x+45$
• $y^2=28 x^6+49 x^5+47 x^4+16 x^3+34 x^2+31 x+30$
• $y^2=51 x^6+28 x^5+45 x^4+42 x^3+45 x^2+28 x+51$
• $y^2=63 x^6+47 x^5+64 x^4+13 x^3+64 x^2+47 x+63$
• $y^2=7 x^6+22 x^5+55 x^4+65 x^3+55 x^2+22 x+7$
• $y^2=46 x^6+12 x^5+9 x^4+27 x^3+42 x^2+35 x+56$
• $y^2=23 x^6+37 x^5+26 x^4+51 x^3+61 x^2+23 x+27$
• $y^2=24 x^6+62 x^5+59 x^4+53 x^3+59 x^2+62 x+24$
• and

• $y^2=9 x^6+51 x^5+53 x^4+21 x^3+53 x^2+51 x+9$
• $y^2=45 x^6+26 x^5+49 x^4+30 x^3+49 x^2+26 x+45$
• $y^2=8 x^6+61 x^5+11 x^4+31 x^3+11 x^2+61 x+8$
• $y^2=11 x^6+49 x^5+35 x^4+53 x^3+65 x^2+12 x+16$
• $y^2=x^6+x^3+25$
• $y^2=x^6+35 x^3+62$
• $y^2=32 x^6+35 x^5+25 x^4+11 x^3+42 x^2+54 x+17$
• $y^2=3 x^6+64 x^5+13 x^4+66 x^3+65 x^2+57 x+41$
• $y^2=31 x^6+3 x^5+24 x^4+64 x^3+24 x^2+3 x+31$
• $y^2=52 x^6+47 x^5+2 x^4+19 x^3+2 x^2+47 x+52$
• $y^2=63 x^6+28 x^5+37 x^4+18 x^3+32 x^2+22 x+20$
• $y^2=14 x^6+19 x^5+16 x^4+40 x^3+46 x^2+59 x+36$
• $y^2=34 x^6+61 x^5+52 x^4+27 x^3+32 x^2+43 x+14$
• $y^2=47 x^6+13 x^5+30 x^4+51 x^3+30 x^2+13 x+47$
• $y^2=4 x^6+11 x^5+55 x^2+26 x+3$
• $y^2=x^6+65 x^3+59$
• $y^2=x^6+19 x^3+25$
• $y^2=6 x^6+40 x^5+55 x^4+19 x^3+31 x^2+10 x+30$
• $y^2=60 x^6+13 x^5+11 x^4+37 x^3+31 x^2+59 x+3$
• $y^2=11 x^6+12 x^5+54 x^4+61 x^3+54 x^2+12 x+11$
• $y^2=55 x^6+58 x^5+12 x^4+48 x^3+12 x^2+58 x+55$
• $y^2=39 x^6+60 x^5+17 x^4+11 x^3+26 x^2+34 x+20$
• $y^2=55 x^6+48 x^5+38 x^4+3 x^3+38 x^2+48 x+55$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{67}$

The isogeny class factors as 1.67.af^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.a_ef	$2$	(not in LMFDB)
2.67.k_gd	$2$	(not in LMFDB)
2.67.aw_jv	$3$	(not in LMFDB)
2.67.aq_hh	$3$	(not in LMFDB)
2.67.f_abq	$3$	(not in LMFDB)
2.67.l_cc	$3$	(not in LMFDB)
2.67.bg_pa	$3$	(not in LMFDB)