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

• Label: 2.67.a_du
• 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:	$( 1 - 6 x + 67 x^{2} )( 1 + 6 x + 67 x^{2} )$
	$1 + 98 x^{2} + 4489 x^{4}$
Frobenius angles:	$\pm0.380553124364$, $\pm0.619446875636$
Angle rank:	$1$ (numerical)
Jacobians:	$56$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4588$	$21049744$	$90458003596$	$406042489046016$	$1822837802339964268$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$68$	$4686$	$300764$	$20149870$	$1350125108$	$90457625022$	$6060711605324$	$406067757377374$	$27206534396294948$	$1822837800128167086$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 56 curves (of which all are hyperelliptic):

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

• $y^2=49 x^6+8 x^5+54 x^4+42 x^3+24 x^2+57 x+38$
• $y^2=31 x^6+16 x^5+41 x^4+17 x^3+48 x^2+47 x+9$
• $y^2=19 x^6+47 x^5+16 x^4+25 x^3+65 x^2+18 x+13$
• $y^2=38 x^6+27 x^5+32 x^4+50 x^3+63 x^2+36 x+26$
• $y^2=58 x^6+55 x^5+19 x^4+12 x^3+52 x^2+36 x+1$
• $y^2=49 x^6+43 x^5+38 x^4+24 x^3+37 x^2+5 x+2$
• $y^2=33 x^6+24 x^5+27 x^4+63 x^3+16 x^2+27 x+5$
• $y^2=66 x^6+48 x^5+54 x^4+59 x^3+32 x^2+54 x+10$
• $y^2=8 x^6+26 x^5+55 x^4+50 x^3+3 x^2+10 x+25$
• $y^2=64 x^6+18 x^5+8 x^4+43 x^3+65 x^2+43 x+66$
• $y^2=15 x^6+4 x^5+12 x^4+33 x^3+24 x^2+16 x+53$
• $y^2=34 x^6+29 x^5+21 x^4+30 x^3+8 x^2+39 x+26$
• $y^2=64 x^6+51 x^5+23 x^4+37 x^3+31 x^2+43 x+42$
• $y^2=19 x^6+40 x^5+31 x^4+37 x^3+65 x^2+39 x+51$
• $y^2=3 x^5+26 x^4+15 x^3+31 x^2+57 x+6$
• $y^2=6 x^5+52 x^4+30 x^3+62 x^2+47 x+12$
• $y^2=53 x^6+17 x^5+28 x^4+13 x^3+37 x^2+39 x+62$
• $y^2=61 x^6+5 x^5+21 x^4+42 x^3+21 x^2+5 x+61$
• $y^2=55 x^6+10 x^5+42 x^4+17 x^3+42 x^2+10 x+55$
• $y^2=29 x^6+55 x^5+57 x^4+33 x^3+23 x^2+33 x+31$
• $y^2=8 x^6+48 x^5+13 x^4+40 x^3+40 x^2+12 x+1$
• $y^2=65 x^6+53 x^5+47 x^4+20 x^3+49 x^2+46$
• $y^2=63 x^6+39 x^5+27 x^4+40 x^3+31 x^2+25$
• $y^2=34 x^6+24 x^5+50 x^4+20 x^3+6 x^2+25 x+39$
• $y^2=x^6+48 x^5+33 x^4+40 x^3+12 x^2+50 x+11$
• $y^2=24 x^6+40 x^5+3 x^4+16 x^3+52 x^2+7 x+58$
• $y^2=48 x^6+13 x^5+6 x^4+32 x^3+37 x^2+14 x+49$
• $y^2=60 x^6+20 x^5+21 x^4+45 x^3+9 x^2+6 x+57$
• $y^2=53 x^6+40 x^5+42 x^4+23 x^3+18 x^2+12 x+47$
• $y^2=49 x^6+25 x^5+59 x^4+57 x^2+60 x+13$
• $y^2=51 x^5+47 x^4+52 x^3+3 x^2+5 x$
• $y^2=18 x^6+24 x^5+7 x^4+22 x^3+66 x^2+5 x+9$
• $y^2=36 x^6+48 x^5+14 x^4+44 x^3+65 x^2+10 x+18$
• $y^2=6 x^6+7 x^5+16 x^4+63 x^3+18 x^2+12 x+2$
• $y^2=48 x^6+28 x^5+31 x^4+26 x^3+59 x^2+x+15$
• $y^2=29 x^6+56 x^5+62 x^4+52 x^3+51 x^2+2 x+30$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{67^{2}}$.

Endomorphism algebra over $\F_{67}$

The isogeny class factors as 1.67.ag $\times$ 1.67.g 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.ag : \(\Q(\sqrt{-58}) \).
• 1.67.g : \(\Q(\sqrt{-58}) \).

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

The base change of $A$ to $\F_{67^{2}}$ is 1.4489.du^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.am_go	$2$	(not in LMFDB)
2.67.m_go	$2$	(not in LMFDB)
2.67.a_adu	$4$	(not in LMFDB)