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

• Label: 2.67.af_abq
• 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:	$( 1 - 16 x + 67 x^{2} )( 1 + 11 x + 67 x^{2} )$
	$1 - 5 x - 42 x^{2} - 335 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.0678686046652$, $\pm0.734535271332$
Angle rank:	$1$ (numerical)
Jacobians:	$33$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4108$	$19669104$	$89930413456$	$406126164540096$	$1822738608157901668$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$63$	$4381$	$299004$	$20154025$	$1350051633$	$90458036422$	$6060715663779$	$406067645681809$	$27206534621379588$	$1822837807250086861$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 33 curves (of which all are hyperelliptic):

• $y^2=25 x^6+60 x^5+32 x^4+44 x^3+36 x^2+43$
• $y^2=8 x^6+42 x^5+38 x^4+37 x^3+13 x^2+16 x+31$
• $y^2=22 x^6+26 x^5+40 x^4+31 x^3+41 x^2+33 x+7$
• $y^2=35 x^6+48 x^5+40 x^4+14 x^3+32 x^2+40 x+18$
• $y^2=60 x^6+63 x^5+44 x^4+18 x^3+42 x^2+39 x+42$
• $y^2=47 x^6+41 x^5+34 x^4+48 x^3+57 x^2+x+58$
• $y^2=58 x^6+2 x^5+42 x^4+10 x^3+53 x^2+33 x+1$
• $y^2=60 x^6+57 x^5+12 x^4+63 x^3+21 x^2+21 x+7$
• $y^2=58 x^6+31 x^5+31 x^4+37 x^3+37 x^2+6 x+6$
• $y^2=36 x^6+7 x^5+20 x^4+42 x^3+54 x^2+24 x+2$
• $y^2=43 x^6+44 x^5+33 x^4+35 x^3+18 x^2+25 x+64$
• $y^2=26 x^6+47 x^5+10 x^4+48 x^3+8 x^2+65 x+15$
• $y^2=16 x^6+3 x^5+63 x^4+65 x^3+7 x^2+10 x$
• $y^2=5 x^6+13 x^5+49 x^4+46 x^3+52 x^2+60 x+6$
• $y^2=34 x^6+65 x^5+31 x^4+7 x^3+50 x^2+4 x+55$
• $y^2=2 x^6+4 x^3+66$
• $y^2=3 x^6+10 x^5+60 x^4+55 x^3+18 x^2+27 x+45$
• $y^2=63 x^6+54 x^5+29 x^3+5 x^2+5 x+64$
• $y^2=56 x^6+16 x^5+31 x^4+6 x^3+15 x^2+15 x+16$
• $y^2=2 x^6+2 x^3+3$
• and

• $y^2=47 x^6+20 x^5+33 x^4+54 x^3+64 x^2+22 x+56$
• $y^2=27 x^6+32 x^5+66 x^4+27 x^3+36 x^2+22 x+19$
• $y^2=8 x^6+31 x^5+14 x^4+44 x^3+30 x^2+47 x+25$
• $y^2=63 x^6+14 x^5+32 x^4+15 x^3+20 x^2+62 x+64$
• $y^2=10 x^6+11 x^5+7 x^4+11 x^3+x^2+31 x+15$
• $y^2=30 x^6+4 x^5+11 x^4+11 x^3+61 x^2+40 x+14$
• $y^2=3 x^6+33 x^5+5 x^4+13 x^3+9 x^2+58 x+3$
• $y^2=36 x^6+46 x^5+8 x^4+8 x^3+7 x^2+65 x+6$
• $y^2=47 x^6+21 x^5+52 x^4+15 x^3+x^2+2 x+37$
• $y^2=2 x^6+2 x^3+63$
• $y^2=2 x^6+2 x^3+38$
• $y^2=11 x^6+24 x^5+19 x^4+35 x^3+33 x^2+27 x+21$
• $y^2=38 x^6+23 x^5+58 x^3+33 x^2+36 x+19$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{67}$

The isogeny class factors as 1.67.aq $\times$ 1.67.l and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.67.aq : \(\Q(\sqrt{-3}) \).
• 1.67.l : \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{67^{3}}$ is 1.300763.abhw^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.abb_ly	$2$	(not in LMFDB)
2.67.f_abq	$2$	(not in LMFDB)
2.67.bb_ly	$2$	(not in LMFDB)
2.67.abg_pa	$3$	(not in LMFDB)
2.67.al_cc	$3$	(not in LMFDB)
2.67.k_gd	$3$	(not in LMFDB)
2.67.q_hh	$3$	(not in LMFDB)
2.67.w_jv	$3$	(not in LMFDB)