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

• Label: 2.67.aba_li
• 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:	$( 1 - 16 x + 67 x^{2} )( 1 - 10 x + 67 x^{2} )$
	$1 - 26 x + 294 x^{2} - 1742 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.0678686046652$, $\pm0.290828956352$
Angle rank:	$2$ (numerical)
Jacobians:	$28$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3016$	$19760832$	$90497194216$	$406106281211904$	$1822804072651158376$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$42$	$4402$	$300894$	$20153038$	$1350100122$	$90457790722$	$6060706327662$	$406067662096606$	$27206534627826378$	$1822837808578152082$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):

• $y^2=50 x^6+15 x^5+39 x^4+x^3+3 x^2+16 x+51$
• $y^2=7 x^6+37 x^5+34 x^4+15 x^3+10 x^2+5 x+34$
• $y^2=10 x^6+59 x^5+13 x^4+5 x^3+13 x^2+59 x+10$
• $y^2=42 x^6+37 x^5+14 x^4+8 x^3+14 x^2+37 x+42$
• $y^2=32 x^5+63 x^4+51 x^3+46 x^2+55 x+63$
• $y^2=14 x^6+4 x^5+10 x^4+15 x^3+47 x^2+50 x+28$
• $y^2=46 x^6+60 x^4+x^3+4 x^2+28 x+50$
• $y^2=41 x^6+14 x^5+7 x^4+58 x^3+7 x^2+14 x+41$
• $y^2=13 x^6+9 x^5+26 x^4+x^2+35 x+28$
• $y^2=31 x^6+57 x^5+48 x^4+49 x^3+8 x^2+63 x+11$
• $y^2=47 x^6+6 x^5+37 x^4+39 x^3+64 x^2+55 x+4$
• $y^2=60 x^6+42 x^5+21 x^4+25 x^3+47 x^2+10 x+2$
• $y^2=29 x^6+50 x^5+50 x^4+14 x^3+54 x^2+22 x+14$
• $y^2=60 x^6+8 x^5+63 x^4+46 x^3+41 x^2+47 x+21$
• $y^2=23 x^6+26 x^5+9 x^4+7 x^3+64 x^2+49 x+17$
• $y^2=21 x^6+47 x^5+50 x^4+58 x^3+31 x^2+65 x+42$
• $y^2=7 x^6+38 x^5+3 x^4+22 x^3+15 x^2+34 x+25$
• $y^2=34 x^5+15 x^4+38 x^3+47 x^2+53 x$
• $y^2=20 x^6+43 x^5+9 x^4+26 x^3+22 x^2+11 x+27$
• $y^2=56 x^6+49 x^5+7 x^4+63 x^3+57 x^2+65 x+22$
• $y^2=58 x^6+6 x^5+9 x^4+40 x^3+52 x^2+45 x+18$
• $y^2=55 x^6+20 x^5+55 x^4+10 x^3+4 x^2+5 x+33$
• $y^2=47 x^6+13 x^5+37 x^4+22 x^3+35 x^2+5 x+27$
• $y^2=53 x^6+7 x^5+52 x^4+2 x^3+54 x^2+13 x+42$
• $y^2=5 x^6+38 x^5+37 x^4+13 x^3+15 x^2+24 x+52$
• $y^2=12 x^6+46 x^5+19 x^4+61 x^3+19 x^2+46 x+12$
• $y^2=17 x^6+38 x^5+49 x^4+12 x^3+26 x^2+11 x+12$
• $y^2=12 x^6+60 x^5+45 x^4+44 x^3+53 x^2+52 x+59$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{67}$

The isogeny class factors as 1.67.aq $\times$ 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.aq : \(\Q(\sqrt{-3}) \).
• 1.67.ak : \(\Q(\sqrt{-42}) \).

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.ag_aba	$2$	(not in LMFDB)
2.67.g_aba	$2$	(not in LMFDB)
2.67.ba_li	$2$	(not in LMFDB)
2.67.af_dg	$3$	(not in LMFDB)
2.67.b_y	$3$	(not in LMFDB)