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

• Label: 2.67.am_go
• 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:	$( 1 - 6 x + 67 x^{2} )^{2}$
	$1 - 12 x + 170 x^{2} - 804 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.380553124364$, $\pm0.380553124364$
Angle rank:	$1$ (numerical)
Jacobians:	$20$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3844$	$21049744$	$91055476516$	$406042489046016$	$1822648562427420484$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$56$	$4686$	$302744$	$20149870$	$1349984936$	$90457625022$	$6060716453960$	$406067757377374$	$27206534550360728$	$1822837800128167086$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

• $y^2=12 x^6+46 x^5+34 x^4+61 x^3+34 x^2+46 x+12$
• $y^2=66 x^6+36 x^5+5 x^4+9 x^3+8 x^2+26 x+18$
• $y^2=57 x^6+39 x^5+35 x^4+50 x^3+14 x^2+18 x+19$
• $y^2=6 x^6+12 x^5+66 x^4+64 x^3+60 x^2+54 x+21$
• $y^2=56 x^6+63 x^5+57 x^4+65 x^3+35 x^2+40 x+27$
• $y^2=22 x^6+x^5+30 x^4+55 x^3+56 x^2+60 x+26$
• $y^2=26 x^6+41 x^5+25 x^4+53 x^3+13 x^2+36 x+26$
• $y^2=28 x^6+2 x^5+52 x^4+42 x^3+56 x^2+66 x+6$
• $y^2=15 x^6+22 x^4+45 x^3+47 x^2+3 x+15$
• $y^2=16 x^6+7 x^5+54 x^4+46 x^3+28 x^2+8 x$
• $y^2=45 x^6+51 x^4+51 x^2+45$
• $y^2=64 x^6+19 x^5+39 x^4+50 x^3+13 x^2+53 x+32$
• $y^2=41 x^6+51 x^5+10 x^4+62 x^3+5 x^2+18 x+24$
• $y^2=9 x^6+18 x^5+49 x^4+10 x^3+20 x^2+34 x+59$
• $y^2=56 x^6+43 x^5+45 x^4+45 x^3+66 x^2+30 x+17$
• $y^2=19 x^6+32 x^5+3 x^4+65 x^3+45 x^2+18 x+38$
• $y^2=62 x^5+45 x^4+63 x^3+52 x^2+46 x+1$
• $y^2=34 x^6+59 x^5+38 x^4+14 x^3+58 x^2+35 x+63$
• $y^2=8 x^6+45 x^5+35 x^4+2 x^3+8 x^2+45 x+31$
• $y^2=27 x^6+42 x^4+16 x^3+58 x^2+27 x+28$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{67}$

The isogeny class factors as 1.67.ag^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-58}) \)$)$

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.a_du	$2$	(not in LMFDB)
2.67.m_go	$2$	(not in LMFDB)
2.67.g_abf	$3$	(not in LMFDB)