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

• Label: 2.67.ai_ft
• 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:	$( 1 - 5 x + 67 x^{2} )( 1 - 3 x + 67 x^{2} )$
	$1 - 8 x + 149 x^{2} - 536 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.401201937998$, $\pm0.441336869475$
Angle rank:	$2$ (numerical)
Jacobians:	$52$
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4095$	$21224385$	$90897402960$	$405875293940025$	$1822659584237466975$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$60$	$4724$	$302220$	$20141572$	$1349993100$	$90458479046$	$6060720394740$	$406067705551108$	$27206533955137140$	$1822837801127574164$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 52 curves (of which all are hyperelliptic):

• $y^2=40 x^6+18 x^5+40 x^4+52 x^3+36 x^2+32 x+64$
• $y^2=63 x^6+47 x^5+66 x^4+x^3+46 x^2+24 x+7$
• $y^2=11 x^6+29 x^5+12 x^4+49 x^3+41 x^2+4 x+46$
• $y^2=25 x^6+21 x^5+33 x^4+39 x^3+54 x^2+59 x+14$
• $y^2=45 x^6+11 x^5+53 x^4+5 x^3+43 x^2+46 x+5$
• $y^2=46 x^6+3 x^5+17 x^4+43 x^3+54 x^2+5 x+11$
• $y^2=35 x^6+17 x^5+17 x^4+16 x^3+59 x^2+55 x+26$
• $y^2=9 x^6+8 x^5+52 x^4+x^3+38 x^2+57 x+62$
• $y^2=55 x^6+46 x^5+38 x^4+42 x^3+12 x^2+7 x+33$
• $y^2=12 x^6+16 x^5+11 x^4+6 x^3+48 x^2+35 x+11$
• $y^2=58 x^6+27 x^5+6 x^4+63 x^3+6 x^2+27 x+58$
• $y^2=51 x^6+4 x^5+17 x^4+45 x^3+59 x^2+9 x+13$
• $y^2=19 x^6+66 x^5+63 x^4+52 x^3+57 x^2+44 x+54$
• $y^2=57 x^6+11 x^5+52 x^4+24 x^3+48 x^2+51 x+57$
• $y^2=52 x^6+23 x^5+24 x^4+56 x^3+26 x^2+47 x+52$
• $y^2=9 x^6+5 x^5+56 x^4+42 x^3+26 x^2+8 x+22$
• $y^2=44 x^6+51 x^5+58 x^4+2 x^3+12 x^2+46 x+57$
• $y^2=8 x^6+26 x^5+19 x^4+31 x^3+37 x^2+29 x+3$
• $y^2=18 x^6+7 x^5+43 x^4+22 x^3+46 x^2+42 x+61$
• $y^2=55 x^6+33 x^5+19 x^4+42 x^3+25 x^2+64 x+4$
• and

• $y^2=66 x^6+38 x^5+27 x^4+58 x^3+32 x^2+27 x+45$
• $y^2=46 x^6+17 x^5+6 x^4+5 x^3+39 x^2+65 x+20$
• $y^2=29 x^6+36 x^5+59 x^4+10 x^3+19 x^2+23 x+33$
• $y^2=44 x^6+41 x^5+35 x^4+60 x^3+24 x^2+44 x+61$
• $y^2=31 x^6+51 x^5+2 x^4+10 x^3+43 x^2+41 x+32$
• $y^2=42 x^6+2 x^5+53 x^4+21 x^3+46 x^2+38 x+58$
• $y^2=33 x^6+36 x^5+23 x^4+21 x^3+6 x^2+35 x+4$
• $y^2=26 x^6+19 x^5+9 x^4+56 x^3+39 x^2+59 x+56$
• $y^2=20 x^6+22 x^5+35 x^4+24 x^3+10 x^2+10 x+20$
• $y^2=64 x^6+62 x^5+11 x^4+27 x^3+63 x^2+x+1$
• $y^2=44 x^6+65 x^5+22 x^4+3 x^3+60 x^2+56 x+57$
• $y^2=24 x^6+5 x^5+61 x^4+5 x^3+5 x^2+50 x+1$
• $y^2=43 x^6+4 x^5+64 x^4+61 x^3+49 x^2+10 x+42$
• $y^2=3 x^6+50 x^5+16 x^4+39 x^3+36 x^2+27 x+8$
• $y^2=2 x^6+39 x^5+34 x^3+35 x+51$
• $y^2=61 x^6+43 x^5+17 x^4+15 x^3+25 x^2+61 x+51$
• $y^2=59 x^6+52 x^5+15 x^4+34 x^3+26 x^2+13 x+1$
• $y^2=45 x^6+23 x^5+33 x^4+17 x^2+56 x+53$
• $y^2=59 x^6+35 x^5+59 x^4+42 x^3+29 x^2+15 x+22$
• $y^2=13 x^6+17 x^5+54 x^4+39 x^3+15 x^2+4 x+28$
• $y^2=25 x^6+31 x^5+25 x^4+40 x^3+60 x^2+66 x+24$
• $y^2=62 x^6+40 x^5+48 x^4+29 x^3+48 x^2+40 x+62$
• $y^2=37 x^6+17 x^5+52 x^4+38 x^3+32 x^2+33 x+6$
• $y^2=63 x^6+4 x^5+55 x^4+40 x^3+65 x^2+15 x+31$
• $y^2=34 x^6+8 x^5+14 x^4+17 x^3+47 x^2+30 x+38$
• $y^2=4 x^6+5 x^5+36 x^4+57 x^3+49 x^2+18 x+33$
• $y^2=13 x^6+3 x^5+31 x^4+12 x^3+34 x^2+43 x+61$
• $y^2=27 x^6+66 x^5+31 x^4+54 x^3+28 x^2+30 x+27$
• $y^2=22 x^6+56 x^5+46 x^4+48 x^3+61 x^2+62 x+22$
• $y^2=61 x^6+15 x^5+39 x^4+62 x^3+6 x^2+40 x+13$
• $y^2=51 x^6+13 x^5+32 x^4+6 x^3+13 x^2+31 x+18$
• $y^2=45 x^6+65 x^5+51 x^4+27 x^3+52 x^2+4 x+58$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{67}$

The isogeny class factors as 1.67.af $\times$ 1.67.ad 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.af : \(\Q(\sqrt{-3}) \).
• 1.67.ad : \(\Q(\sqrt{-259}) \).

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.ac_ep	$2$	(not in LMFDB)
2.67.c_ep	$2$	(not in LMFDB)
2.67.i_ft	$2$	(not in LMFDB)
2.67.ao_gl	$3$	(not in LMFDB)
2.67.n_di	$3$	(not in LMFDB)