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

• Label: 2.67.aba_lr
• 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:	$( 1 - 13 x + 67 x^{2} )^{2}$
	$1 - 26 x + 303 x^{2} - 1742 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.207941879321$, $\pm0.207941879321$
Angle rank:	$1$ (numerical)
Jacobians:	$16$
Cyclic group of points:	no
Non-cyclic primes:	$5, 11$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3025$	$19847025$	$90709392400$	$406380241265625$	$1823034706713900625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$42$	$4420$	$301596$	$20166628$	$1350270942$	$90459239110$	$6060712974666$	$406067637943108$	$27206533789573092$	$1822837799318484100$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):

• $y^2=x^6+14 x^5+14 x^4+36 x^3+14 x^2+14 x+1$
• $y^2=63 x^6+64 x^5+37 x^4+49 x^3+20 x^2+27 x+54$
• $y^2=2 x^6+7 x^3+2$
• $y^2=x^6+18 x^5+63 x^4+59 x^3+5 x^2+18 x+4$
• $y^2=31 x^6+7 x^5+35 x^4+21 x^3+35 x^2+7 x+31$
• $y^2=45 x^6+x^5+24 x^4+63 x^3+24 x^2+x+45$
• $y^2=26 x^6+53 x^5+41 x^4+15 x^3+25 x^2+62 x+39$
• $y^2=18 x^6+38 x^5+54 x^4+30 x^3+54 x^2+38 x+18$
• $y^2=7 x^6+26 x^5+16 x^4+14 x^3+16 x^2+26 x+7$
• $y^2=2 x^6+2 x^3+51$
• $y^2=50 x^6+56 x^5+40 x^4+65 x^3+40 x^2+56 x+50$
• $y^2=41 x^6+55 x^5+6 x^4+13 x^3+6 x^2+55 x+41$
• $y^2=21 x^6+24 x^5+42 x^4+38 x^3+42 x^2+24 x+21$
• $y^2=18 x^6+39 x^5+42 x^4+x^3+42 x^2+39 x+18$
• $y^2=2 x^6+2 x^3+48$
• $y^2=66 x^6+3 x^5+14 x^4+29 x^3+19 x^2+38 x+57$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{67}$

The isogeny class factors as 1.67.an^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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_abj	$2$	(not in LMFDB)
2.67.ba_lr	$2$	(not in LMFDB)
2.67.n_dy	$3$	(not in LMFDB)