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

• Label: 2.67.l_gg
• 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:	$( 1 + 4 x + 67 x^{2} )( 1 + 7 x + 67 x^{2} )$
	$1 + 11 x + 162 x^{2} + 737 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.578570930462$, $\pm0.640638367129$
Angle rank:	$2$ (numerical)
Jacobians:	$48$
Isomorphism classes:	200

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5400$	$21081600$	$89917192800$	$406003366656000$	$1823011141757157000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$79$	$4693$	$298960$	$20147929$	$1350253489$	$90457905526$	$6060706502083$	$406067730625201$	$27206534414250160$	$1822837801657891093$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 48 curves (of which all are hyperelliptic):

• $y^2=6 x^6+39 x^5+47 x^4+22 x^3+7 x^2+16 x+18$
• $y^2=62 x^6+65 x^5+10 x^4+25 x^3+58 x^2+65 x+30$
• $y^2=6 x^6+9 x^5+x^4+42 x^3+3 x^2+20 x+29$
• $y^2=45 x^6+26 x^5+21 x^4+2 x^3+38 x^2+30 x+7$
• $y^2=28 x^6+55 x^5+34 x^4+23 x^3+x^2+22 x+24$
• $y^2=10 x^6+41 x^5+64 x^4+54 x^3+42 x^2+49 x+16$
• $y^2=54 x^6+19 x^5+34 x^4+36 x^3+43 x^2+49 x+59$
• $y^2=34 x^6+56 x^5+44 x^4+58 x^3+65 x^2+36 x+22$
• $y^2=60 x^6+42 x^5+20 x^4+48 x^3+57 x^2+55 x$
• $y^2=46 x^6+47 x^5+10 x^4+43 x^3+21 x^2+31 x+39$
• $y^2=49 x^6+9 x^5+49 x^4+43 x^3+9 x^2+2 x+59$
• $y^2=45 x^6+15 x^5+31 x^4+32 x^3+49 x^2+51 x+49$
• $y^2=29 x^5+16 x^4+24 x^3+28 x^2+41 x+33$
• $y^2=18 x^6+x^5+59 x^4+60 x^3+12 x^2+6 x+13$
• $y^2=58 x^6+63 x^5+28 x^4+58 x^3+34 x^2+18 x+18$
• $y^2=60 x^6+65 x^5+3 x^4+3 x^3+24 x^2+57 x+65$
• $y^2=65 x^6+14 x^5+62 x^4+44 x^3+30 x^2+19 x+13$
• $y^2=28 x^6+46 x^5+11 x^4+14 x^3+28 x^2+52 x+41$
• $y^2=20 x^6+58 x^5+52 x^4+19 x^3+31 x^2+43 x$
• $y^2=27 x^6+38 x^5+14 x^4+23 x^3+19 x^2+34 x+9$
• and

• $y^2=45 x^6+49 x^5+23 x^4+15 x^3+61 x^2+4 x+42$
• $y^2=4 x^6+38 x^5+23 x^4+53 x^3+56 x^2+47 x+33$
• $y^2=48 x^6+44 x^5+36 x^4+10 x^3+3 x^2+64$
• $y^2=55 x^6+50 x^5+21 x^4+48 x^3+31 x^2+46 x+53$
• $y^2=10 x^6+47 x^5+26 x^4+10 x^3+55 x^2+16 x+28$
• $y^2=7 x^6+39 x^5+21 x^4+32 x^3+21 x+61$
• $y^2=12 x^6+3 x^5+46 x^4+6 x^3+18 x^2+21 x+11$
• $y^2=55 x^6+58 x^5+52 x^4+33 x^3+66 x^2+53 x+4$
• $y^2=8 x^6+19 x^5+53 x^4+20 x^3+57 x^2+61 x+36$
• $y^2=30 x^6+43 x^5+16 x^4+13 x^3+41 x^2+60 x+26$
• $y^2=26 x^6+49 x^5+31 x^4+16 x^3+47 x^2+55 x+33$
• $y^2=x^6+57 x^5+63 x^3+62 x^2+9 x+44$
• $y^2=6 x^6+41 x^5+58 x^4+16 x^3+23 x^2+26 x+6$
• $y^2=17 x^6+64 x^5+2 x^4+63 x^3+63 x^2+65 x+9$
• $y^2=37 x^6+11 x^5+63 x^4+20 x^3+60 x^2+49 x+64$
• $y^2=13 x^6+27 x^5+48 x^4+18 x^3+47 x^2+28 x+1$
• $y^2=61 x^6+57 x^5+39 x^4+61 x^3+53 x^2+39 x+60$
• $y^2=24 x^6+30 x^5+3 x^4+31 x^3+6 x^2+55 x+54$
• $y^2=11 x^6+28 x^5+5 x^4+22 x^3+21 x^2+60 x+64$
• $y^2=58 x^6+58 x^5+11 x^4+41 x^3+53 x^2+61 x+8$
• $y^2=61 x^6+53 x^5+46 x^4+47 x^3+33 x^2+25 x+66$
• $y^2=63 x^6+34 x^5+8 x^4+62 x^3+15 x^2+13 x+12$
• $y^2=2 x^6+64 x^5+13 x^4+57 x^3+49 x^2+4 x+29$
• $y^2=7 x^6+45 x^5+56 x^4+46 x^3+30 x^2+65 x+59$
• $y^2=29 x^6+34 x^5+28 x^4+53 x^3+24 x^2+53 x+18$
• $y^2=48 x^6+12 x^5+22 x^4+7 x^3+8 x^2+51 x+34$
• $y^2=42 x^6+64 x^5+62 x^4+58 x^3+66 x^2+63 x+21$
• $y^2=6 x^6+63 x^5+46 x^4+38 x^3+37 x^2+12 x+53$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{67}$

The isogeny class factors as 1.67.e $\times$ 1.67.h 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.e : \(\Q(\sqrt{-7}) \).
• 1.67.h : \(\Q(\sqrt{-219}) \).

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.al_gg	$2$	(not in LMFDB)
2.67.ad_ec	$2$	(not in LMFDB)
2.67.d_ec	$2$	(not in LMFDB)