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

• Label: 2.67.a_aeg
• 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:	$1 - 110 x^{2} + 4489 x^{4}$
Frobenius angles:	$\pm0.0967373897055$, $\pm0.903262610295$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-6}, \sqrt{61})\)
Galois group:	$C_2^2$
Jacobians:	$66$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4380$	$19184400$	$90458532540$	$405941904000000$	$1822837807237839900$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$68$	$4270$	$300764$	$20144878$	$1350125108$	$90458682910$	$6060711605324$	$406067738667358$	$27206534396294948$	$1822837809923918350$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 66 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{67}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-6}, \sqrt{61})\).

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

The base change of $A$ to $\F_{67^{2}}$ is 1.4489.aeg^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-366}) \)$)$

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.a_eg	$4$	(not in LMFDB)