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:	$28$
Isomorphism classes:	112

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 contains the Jacobians of 28 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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)