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

• Label: 2.67.g_fm
• 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:	$( 1 + 2 x + 67 x^{2} )( 1 + 4 x + 67 x^{2} )$
	$1 + 6 x + 142 x^{2} + 402 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.538985133153$, $\pm0.578570930462$
Angle rank:	$2$ (numerical)
Jacobians:	$80$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5040$	$21288960$	$90118208880$	$405808452403200$	$1822988492244889200$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$74$	$4738$	$299630$	$20138254$	$1350236714$	$90458882386$	$6060703019438$	$406067670940126$	$27206534953103690$	$1822837803356510818$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 80 curves (of which all are hyperelliptic):

• $y^2=x^6+23 x^5+10 x^4+14 x^3+62 x^2+56 x+25$
• $y^2=58 x^6+18 x^5+32 x^4+7 x^3+32 x^2+18 x+58$
• $y^2=32 x^6+10 x^5+58 x^4+36 x^3+58 x^2+10 x+32$
• $y^2=x^6+52 x^5+10 x^4+51 x^3+4 x^2+11 x+14$
• $y^2=4 x^6+42 x^5+66 x^4+52 x^3+66 x^2+42 x+4$
• $y^2=9 x^6+48 x^5+2 x^4+22 x^3+58 x^2+34 x+9$
• $y^2=30 x^6+39 x^5+21 x^4+44 x^3+21 x^2+39 x+30$
• $y^2=55 x^6+21 x^5+15 x^4+48 x^3+15 x^2+21 x+55$
• $y^2=36 x^6+33 x^5+16 x^4+20 x^3+16 x^2+33 x+36$
• $y^2=31 x^6+44 x^5+36 x^4+x^3+36 x^2+44 x+31$
• $y^2=47 x^6+2 x^5+17 x^4+48 x^3+17 x^2+2 x+47$
• $y^2=62 x^6+37 x^5+37 x^4+47 x^3+37 x^2+37 x+62$
• $y^2=37 x^6+11 x^5+36 x^4+42 x^3+22 x^2+13 x+23$
• $y^2=43 x^6+23 x^5+26 x^4+39 x^3+26 x^2+23 x+43$
• $y^2=53 x^6+57 x^5+33 x^4+47 x^3+33 x^2+57 x+53$
• $y^2=54 x^6+55 x^5+9 x^4+56 x^3+9 x^2+55 x+54$
• $y^2=26 x^6+10 x^5+59 x^3+64 x+35$
• $y^2=12 x^6+34 x^5+57 x^4+4 x^3+57 x^2+34 x+12$
• $y^2=48 x^6+29 x^5+12 x^4+26 x^3+43 x^2+49 x+18$
• $y^2=28 x^6+33 x^5+2 x^4+33 x^3+58 x^2+15 x+28$
• and

• $y^2=65 x^6+61 x^5+5 x^4+17 x^3+5 x^2+61 x+65$
• $y^2=53 x^6+43 x^5+18 x^4+17 x^3+18 x^2+43 x+53$
• $y^2=37 x^6+26 x^5+40 x^4+57 x^3+40 x^2+26 x+37$
• $y^2=49 x^6+35 x^5+50 x^4+25 x^3+8 x^2+62 x+23$
• $y^2=16 x^6+39 x^5+61 x^4+21 x^3+61 x^2+39 x+16$
• $y^2=11 x^6+55 x^5+38 x^4+17 x^3+3 x^2+17 x+38$
• $y^2=40 x^6+4 x^5+12 x^4+25 x^3+12 x^2+4 x+40$
• $y^2=64 x^6+37 x^5+31 x^4+44 x^3+31 x^2+37 x+64$
• $y^2=27 x^6+22 x^5+64 x^4+48 x^3+26 x^2+37 x+45$
• $y^2=51 x^6+10 x^5+7 x^3+14 x+57$
• $y^2=6 x^6+38 x^5+25 x^4+32 x^3+17 x^2+18 x+19$
• $y^2=3 x^6+61 x^5+38 x^4+30 x^3+63 x^2+30 x+58$
• $y^2=29 x^6+26 x^5+37 x^4+21 x^3+23 x^2+33 x+21$
• $y^2=6 x^6+66 x^5+59 x^4+53 x^3+49 x^2+41 x+16$
• $y^2=41 x^6+15 x^5+57 x^4+35 x^3+44 x^2+9 x+7$
• $y^2=51 x^6+56 x^5+59 x^4+49 x^3+59 x^2+56 x+51$
• $y^2=65 x^6+18 x^5+61 x^4+44 x^3+61 x^2+18 x+65$
• $y^2=7 x^6+16 x^5+12 x^4+40 x^3+12 x^2+16 x+7$
• $y^2=30 x^6+32 x^5+28 x^4+61 x^3+28 x^2+32 x+30$
• $y^2=42 x^6+57 x^5+66 x^4+47 x^3+46 x^2+12 x+27$
• $y^2=64 x^6+43 x^5+27 x^4+20 x^3+27 x^2+43 x+64$
• $y^2=20 x^6+5 x^5+36 x^4+12 x^3+9 x^2+38 x+38$
• $y^2=28 x^6+65 x^5+34 x^4+60 x^3+34 x^2+65 x+28$
• $y^2=27 x^6+38 x^4+41 x^3+52 x^2+5$
• $y^2=37 x^6+36 x^5+21 x^4+47 x^3+21 x^2+36 x+37$
• $y^2=39 x^6+9 x^5+43 x^4+56 x^3+13 x^2+33 x+37$
• $y^2=25 x^6+56 x^5+51 x^4+13 x^3+61 x^2+33 x+64$
• $y^2=28 x^6+58 x^5+29 x^4+24 x^3+9 x^2+31 x+44$
• $y^2=46 x^6+59 x^5+36 x^4+35 x^3+36 x^2+59 x+46$
• $y^2=55 x^6+65 x^5+32 x^4+65 x^3+46 x^2+10 x+36$
• $y^2=58 x^6+26 x^5+4 x^4+20 x^3+4 x^2+26 x+58$
• $y^2=3 x^6+22 x^5+61 x^4+9 x^3+13 x^2+40 x+43$
• $y^2=39 x^6+5 x^4+13 x^3+28 x^2+37$
• $y^2=66 x^6+5 x^5+50 x^4+6 x^3+61 x^2+66 x+45$
• $y^2=56 x^6+55 x^5+63 x^4+9 x^3+63 x^2+55 x+56$
• $y^2=22 x^6+57 x^5+57 x^4+36 x^3+57 x^2+57 x+22$
• $y^2=9 x^6+10 x^5+46 x^4+48 x^3+46 x^2+10 x+9$
• $y^2=37 x^6+10 x^5+35 x^4+52 x^3+35 x^2+10 x+37$
• $y^2=48 x^6+55 x^5+41 x^4+22 x^3+66 x^2+65 x+57$
• $y^2=7 x^6+46 x^5+34 x^4+2 x^3+34 x^2+46 x+7$
• $y^2=44 x^6+6 x^5+62 x^4+48 x^3+62 x^2+6 x+44$
• $y^2=20 x^6+17 x^5+15 x^4+50 x^3+62 x^2+54 x+34$
• $y^2=28 x^6+45 x^5+2 x^4+15 x^3+2 x^2+45 x+28$
• $y^2=25 x^6+32 x^5+50 x^4+39 x^3+50 x^2+32 x+25$
• $y^2=62 x^6+58 x^5+45 x^4+45 x^3+45 x^2+58 x+62$
• $y^2=41 x^6+25 x^5+45 x^4+10 x^3+50 x^2+35 x+34$
• $y^2=44 x^6+16 x^5+41 x^4+39 x^3+43 x^2+62 x+44$
• $y^2=35 x^6+55 x^5+23 x^4+14 x^3+37 x^2+21 x+26$
• $y^2=5 x^6+55 x^5+22 x^4+17 x^3+22 x^2+55 x+5$
• $y^2=43 x^6+66 x^5+52 x^4+41 x^3+52 x^2+66 x+43$
• $y^2=49 x^6+46 x^5+17 x^4+x^3+47 x^2+2 x+16$
• $y^2=x^6+33 x^5+10 x^4+43 x^3+4 x^2+16 x+14$
• $y^2=57 x^6+48 x^5+x^4+32 x^3+39 x^2+45 x+28$
• $y^2=2 x^6+43 x^5+34 x^4+29 x^3+38 x^2+66 x+51$
• $y^2=62 x^6+63 x^5+49 x^4+16 x^3+49 x^2+63 x+62$
• $y^2=30 x^6+14 x^5+30 x^4+4 x^3+48 x^2+x+13$
• $y^2=58 x^6+29 x^5+51 x^4+58 x^3+51 x^2+29 x+58$
• $y^2=33 x^6+2 x^5+12 x^4+4 x^3+12 x^2+2 x+33$
• $y^2=19 x^6+49 x^5+18 x^4+51 x^3+18 x^2+49 x+19$
• $y^2=59 x^6+45 x^5+58 x^3+11 x+40$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{67}$

The isogeny class factors as 1.67.c $\times$ 1.67.e 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.c : \(\Q(\sqrt{-66}) \).
• 1.67.e : \(\Q(\sqrt{-7}) \).

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.ag_fm	$2$	(not in LMFDB)
2.67.ac_ew	$2$	(not in LMFDB)
2.67.c_ew	$2$	(not in LMFDB)