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

• Label: 2.67.ac_acl
• Base field: $\F_{67}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{67}$
Dimension:	$2$
L-polynomial:	$1 - 2 x - 63 x^{2} - 134 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.127681533513$, $\pm0.794348200180$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{22})\)
Galois group:	$C_2^2$
Jacobians:	$40$
Cyclic group of points:	yes

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4291$	$19579833$	$90222136900$	$406227357352089$	$1822780773001210771$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$66$	$4360$	$299976$	$20159044$	$1350082866$	$90459274750$	$6060715328118$	$406067700012484$	$27206534984972712$	$1822837803635897800$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 40 curves (of which all are hyperelliptic):

• $y^2=8 x^6+63 x^5+43 x^4+23 x^3+7 x^2+59 x+39$
• $y^2=63 x^6+16 x^5+60 x^4+18 x^3+50 x^2+34 x+56$
• $y^2=32 x^6+3 x^5+13 x^4+44 x^3+22 x^2+53 x+14$
• $y^2=53 x^6+34 x^5+11 x^4+43 x^3+12 x^2+36 x+43$
• $y^2=2 x^6+4 x^3+53$
• $y^2=37 x^6+22 x^5+44 x^4+19 x^3+15 x^2+43 x+58$
• $y^2=62 x^6+25 x^5+35 x^4+29 x^3+54 x^2+49 x+55$
• $y^2=8 x^6+14 x^5+8 x^4+15 x^3+3 x^2+53 x+32$
• $y^2=10 x^6+17 x^5+31 x^4+59 x^3+50 x^2+13 x+6$
• $y^2=46 x^6+44 x^5+60 x^4+9 x^3+3 x^2+17 x+20$
• $y^2=2 x^6+2 x^3+42$
• $y^2=7 x^6+12 x^5+10 x^4+18 x^3+51 x^2+63 x+34$
• $y^2=58 x^6+23 x^5+8 x^4+52 x^3+56 x^2+22 x+54$
• $y^2=15 x^6+37 x^5+42 x^4+25 x^3+57 x^2+42 x+3$
• $y^2=4 x^6+39 x^5+30 x^4+49 x^3+13 x^2+53 x+33$
• $y^2=56 x^6+27 x^5+48 x^4+55 x^3+66 x^2+43 x+8$
• $y^2=7 x^6+22 x^5+27 x^4+37 x^3+48 x^2+5 x+39$
• $y^2=22 x^6+57 x^5+19 x^4+43 x^3+61 x^2+50 x+4$
• $y^2=62 x^6+62 x^5+42 x^4+10 x^3+18 x^2+30 x+16$
• $y^2=17 x^6+24 x^5+56 x^4+22 x^3+56 x^2+4 x+8$
• and

• $y^2=34 x^6+34 x^5+14 x^4+17 x^3+32 x^2+52 x+24$
• $y^2=27 x^6+55 x^5+4 x^4+42 x^3+11 x^2+4 x+21$
• $y^2=4 x^6+61 x^5+12 x^4+17 x^3+29 x^2+11 x+57$
• $y^2=28 x^6+24 x^5+8 x^4+30 x^3+64 x^2+64 x+17$
• $y^2=30 x^6+35 x^5+24 x^4+44 x^3+24 x^2+18 x+14$
• $y^2=24 x^6+20 x^5+8 x^3+66 x^2+65 x+2$
• $y^2=35 x^6+43 x^5+2 x^4+7 x^3+33 x^2+18 x+38$
• $y^2=2 x^6+4 x^3+45$
• $y^2=39 x^6+2 x^5+22 x^4+60 x^3+17 x^2+41 x+60$
• $y^2=59 x^6+20 x^5+58 x^4+46 x^3+44 x^2+23 x+49$
• $y^2=22 x^6+60 x^5+64 x^4+26 x^3+37 x^2+5 x+48$
• $y^2=38 x^6+28 x^5+4 x^4+15 x^3+48 x^2+3 x+13$
• $y^2=2 x^6+2 x^3+52$
• $y^2=23 x^6+38 x^5+54 x^4+19 x^3+37 x^2+39 x+42$
• $y^2=46 x^6+48 x^5+7 x^4+29 x^3+21 x^2+25 x+34$
• $y^2=26 x^6+54 x^5+14 x^4+46 x^3+17 x^2+27 x+16$
• $y^2=17 x^6+39 x^5+11 x^4+47 x^3+66 x^2+3 x+12$
• $y^2=28 x^6+31 x^5+16 x^4+62 x^3+52 x^2+59 x+4$
• $y^2=48 x^6+60 x^5+40 x^4+61 x^3+58 x^2+3 x+51$
• $y^2=65 x^6+60 x^5+66 x^4+58 x^3+24 x^2+55 x+19$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{67^{3}}$.

Endomorphism algebra over $\F_{67}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{22})\).

Endomorphism algebra over $\overline{\F}_{67}$

The base change of $A$ to $\F_{67^{3}}$ is 1.300763.ape^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-66}) \)$)$

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.c_acl	$2$	(not in LMFDB)
2.67.e_fi	$3$	(not in LMFDB)
2.67.ae_fi	$6$	(not in LMFDB)