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

• Label: 2.67.u_ja
• 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:	$( 1 + 10 x + 67 x^{2} )^{2}$
	$1 + 20 x + 234 x^{2} + 1340 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.709171043648$, $\pm0.709171043648$
Angle rank:	$1$ (numerical)
Jacobians:	$32$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6084$	$20466576$	$89852460516$	$406383023195136$	$1822809319723535364$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$88$	$4558$	$298744$	$20166766$	$1350104008$	$90457545022$	$6060721390504$	$406067635793758$	$27206534158316728$	$1822837809729656878$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 32 curves (of which all are hyperelliptic):

• $y^2=33 x^6+62 x^5+13 x^4+26 x^3+13 x^2+62 x+33$
• $y^2=50 x^6+56 x^5+38 x^4+56 x^3+38 x^2+56 x+50$
• $y^2=26 x^6+59 x^5+11 x^4+10 x^3+63 x^2+15 x+36$
• $y^2=63 x^6+x^5+57 x^4+63 x^3+57 x^2+x+63$
• $y^2=25 x^6+25 x^4+25 x^2+25$
• $y^2=27 x^6+27 x^5+35 x^4+10 x^3+14 x^2+7 x+43$
• $y^2=38 x^6+19 x^5+15 x^4+4 x^3+x^2+22 x+4$
• $y^2=9 x^6+23 x^5+14 x^4+33 x^3+59 x^2+19 x+17$
• $y^2=43 x^6+21 x^5+56 x^4+50 x^3+4 x^2+36 x+8$
• $y^2=19 x^6+46 x^5+5 x^4+8 x^3+63 x^2+6 x+15$
• $y^2=31 x^6+41 x^5+27 x^4+22 x^3+45 x^2+27 x+40$
• $y^2=54 x^6+10 x^5+47 x^4+4 x^3+33 x^2+6 x+24$
• $y^2=47 x^6+32 x^5+28 x^4+28 x^3+11 x^2+56 x+54$
• $y^2=56 x^5+65 x^4+2 x^3+59 x^2+58 x+36$
• $y^2=21 x^6+21 x^5+47 x^4+x^3+12 x^2+46 x+15$
• $y^2=32 x^6+49 x^4+49 x^2+32$
• $y^2=12 x^6+2 x^5+x^4+46 x^3+x^2+2 x+12$
• $y^2=48 x^6+6 x^5+8 x^4+3 x^3+8 x^2+6 x+48$
• $y^2=56 x^6+58 x^5+43 x^4+54 x^3+43 x^2+6 x+9$
• $y^2=13 x^6+17 x^5+48 x^4+66 x^3+10 x^2+6 x+9$
• and

• $y^2=65 x^6+64 x^5+59 x^4+37 x^3+14 x^2+62 x+17$
• $y^2=33 x^6+21 x^5+41 x^4+37 x^3+65 x^2+35 x+6$
• $y^2=x^6+3 x^5+53 x^4+16 x^3+20 x^2+43 x+60$
• $y^2=6 x^6+32 x^5+28 x^4+46 x^3+26 x^2+40 x+23$
• $y^2=42 x^6+62 x^5+35 x^4+22 x^3+60 x^2+17 x+2$
• $y^2=x^6+33 x^3+25$
• $y^2=17 x^6+63 x^5+57 x^4+44 x^3+65 x^2+60 x+53$
• $y^2=54 x^6+55 x^5+42 x^4+47 x^3+61 x^2+63 x$
• $y^2=14 x^6+5 x^5+14 x^4+32 x^3+14 x^2+5 x+14$
• $y^2=5 x^5+28 x^4+13 x^3+39 x^2+18 x+23$
• $y^2=57 x^6+6 x^5+14 x^4+46 x^3+35 x^2+12 x+4$
• $y^2=x^6+39 x^3+24$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{67}$

The isogeny class factors as 1.67.k^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-42}) \)$)$

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.au_ja	$2$	(not in LMFDB)
2.67.a_bi	$2$	(not in LMFDB)
2.67.ak_bh	$3$	(not in LMFDB)