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

• Label: 2.67.aw_ju
• 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:	$( 1 - 12 x + 67 x^{2} )( 1 - 10 x + 67 x^{2} )$
	$1 - 22 x + 254 x^{2} - 1474 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.238111713333$, $\pm0.290828956352$
Angle rank:	$2$ (numerical)
Jacobians:	$24$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3248$	$20267520$	$90969168752$	$406404311040000$	$1822934015107426928$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$46$	$4514$	$302458$	$20167822$	$1350196366$	$90458097266$	$6060704249338$	$406067618158558$	$27206534218131886$	$1822837806155498114$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):

• $y^2=14 x^6+5 x^5+12 x^4+61 x^3+12 x^2+5 x+14$
• $y^2=44 x^6+63 x^5+48 x^4+59 x^3+8 x^2+52 x+61$
• $y^2=55 x^6+39 x^5+50 x^4+40 x^3+61 x^2+19 x+4$
• $y^2=5 x^6+48 x^5+59 x^4+62 x^3+35 x^2+31 x+52$
• $y^2=41 x^6+40 x^5+60 x^4+6 x^3+64 x^2+47 x+63$
• $y^2=44 x^6+24 x^4+10 x^3+24 x^2+44$
• $y^2=24 x^6+24 x^5+8 x^4+49 x^3+44 x^2+56 x+40$
• $y^2=59 x^6+15 x^5+15 x^4+14 x^3+22 x^2+x+15$
• $y^2=3 x^6+3 x^5+57 x^4+29 x^3+57 x^2+3 x+3$
• $y^2=41 x^6+17 x^5+65 x^4+66 x^3+65 x^2+17 x+41$
• $y^2=27 x^6+54 x^5+54 x^4+30 x^3+54 x^2+54 x+27$
• $y^2=17 x^6+54 x^5+9 x^4+47 x^3+9 x^2+54 x+17$
• $y^2=2 x^6+50 x^5+61 x^4+33 x^3+61 x^2+50 x+2$
• $y^2=64 x^6+32 x^5+62 x^4+13 x^3+62 x^2+32 x+64$
• $y^2=3 x^6+20 x^5+15 x^4+45 x^3+36 x^2+8 x+27$
• $y^2=46 x^6+55 x^5+2 x^4+45 x^3+53 x^2+15 x+34$
• $y^2=61 x^6+48 x^5+35 x^4+27 x^3+35 x^2+48 x+61$
• $y^2=32 x^6+47 x^5+10 x^4+46 x^3+10 x^2+47 x+32$
• $y^2=34 x^6+46 x^5+50 x^4+14 x^3+46 x^2+28 x+11$
• $y^2=20 x^6+23 x^5+14 x^4+41 x^3+14 x^2+23 x+20$
• $y^2=5 x^6+11 x^5+63 x^4+59 x^3+5 x^2+13 x+52$
• $y^2=31 x^6+16 x^5+27 x^4+33 x^3+27 x^2+16 x+31$
• $y^2=43 x^6+32 x^5+10 x^4+6 x^3+10 x^2+32 x+43$
• $y^2=19 x^6+26 x^5+51 x^4+38 x^3+51 x^2+26 x+19$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{67}$

The isogeny class factors as 1.67.am $\times$ 1.67.ak 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.am : \(\Q(\sqrt{-31}) \).
• 1.67.ak : \(\Q(\sqrt{-42}) \).

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_o	$2$	(not in LMFDB)
2.67.c_o	$2$	(not in LMFDB)
2.67.w_ju	$2$	(not in LMFDB)