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

• Label: 2.67.aba_lq
• 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:	$( 1 - 14 x + 67 x^{2} )( 1 - 12 x + 67 x^{2} )$
	$1 - 26 x + 302 x^{2} - 1742 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.173442769152$, $\pm0.238111713333$
Angle rank:	$2$ (numerical)
Jacobians:	$18$
Isomorphism classes:	72

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3024$	$19837440$	$90685807632$	$406350120960000$	$1823010484231301904$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$42$	$4418$	$301518$	$20165134$	$1350253002$	$90459112466$	$6060712993230$	$406067652984286$	$27206534036325546$	$1822837801752842018$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

• $y^2=44 x^6+3 x^5+32 x^4+4 x^3+31 x^2+53 x+30$
• $y^2=40 x^6+49 x^5+52 x^4+23 x^3+52 x^2+49 x+40$
• $y^2=53 x^6+59 x^5+6 x^4+7 x^3+6 x^2+59 x+53$
• $y^2=42 x^6+50 x^5+27 x^4+46 x^3+31 x^2+7 x+27$
• $y^2=7 x^6+17 x^5+55 x^4+35 x^3+54 x^2+26 x+7$
• $y^2=22 x^6+56 x^5+10 x^4+30 x^3+14 x^2+24 x+40$
• $y^2=61 x^6+50 x^5+11 x^4+20 x^3+11 x^2+50 x+61$
• $y^2=61 x^6+46 x^5+33 x^4+58 x^3+25 x^2+28 x+2$
• $y^2=28 x^6+21 x^4+52 x^3+21 x^2+28$
• $y^2=57 x^6+53 x^5+5 x^4+36 x^3+20 x^2+44 x+30$
• $y^2=38 x^6+64 x^5+22 x^4+17 x^3+22 x^2+64 x+38$
• $y^2=66 x^6+7 x^5+58 x^4+30 x^3+58 x^2+7 x+66$
• $y^2=5 x^6+56 x^5+45 x^4+23 x^3+7 x^2+40 x+8$
• $y^2=30 x^6+50 x^5+48 x^4+36 x^3+66 x^2+38 x+28$
• $y^2=43 x^6+62 x^5+4 x^4+59 x^3+4 x^2+62 x+43$
• $y^2=3 x^6+29 x^5+29 x^4+57 x^3+39 x^2+64 x+5$
• $y^2=5 x^6+13 x^5+37 x^4+44 x^3+37 x^2+13 x+5$
• $y^2=42 x^6+20 x^5+61 x^4+48 x^3+61 x^2+20 x+42$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{67}$

The isogeny class factors as 1.67.ao $\times$ 1.67.am 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.ao : \(\Q(\sqrt{-2}) \).
• 1.67.am : \(\Q(\sqrt{-31}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.67.ac_abi	$2$	(not in LMFDB)
2.67.c_abi	$2$	(not in LMFDB)
2.67.ba_lq	$2$	(not in LMFDB)