Abelian variety isogeny class 2.79.a_afk over $\F_{79}$

• Label: 2.79.a_afk
• Base field: $\F_{79}$
• 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:	$\F_{79}$
Dimension:	$2$
L-polynomial:	$1 - 140 x^{2} + 6241 x^{4}$
Frobenius angles:	$\pm0.0767104994935$, $\pm0.923289500506$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}, \sqrt{-149})\)
Galois group:	$C_2^2$
Jacobians:	$42$
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6102$	$37234404$	$243087332742$	$1516554445105296$	$9468276087205910502$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$80$	$5962$	$493040$	$38935846$	$3077056400$	$243087209962$	$19203908986160$	$1517108864375038$	$119851595982618320$	$9468276091784973802$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 42 curves (of which all are hyperelliptic):

• $y^2=69 x^6+47 x^5+27 x^4+68 x^3+17 x^2+23 x+51$
• $y^2=49 x^6+62 x^5+2 x^4+46 x^3+51 x^2+69 x+74$
• $y^2=39 x^6+11 x^5+53 x^4+59 x^3+24 x^2+58 x+77$
• $y^2=38 x^6+33 x^5+x^4+19 x^3+72 x^2+16 x+73$
• $y^2=43 x^6+68 x^5+27 x^4+x^3+42 x^2+13 x+62$
• $y^2=50 x^6+46 x^5+2 x^4+3 x^3+47 x^2+39 x+28$
• $y^2=62 x^6+41 x^5+53 x^4+67 x^3+7 x^2+61 x+40$
• $y^2=28 x^6+44 x^5+x^4+43 x^3+21 x^2+25 x+41$
• $y^2=50 x^6+15 x^5+7 x^4+64 x^3+10 x^2+21 x+53$
• $y^2=71 x^6+45 x^5+21 x^4+34 x^3+30 x^2+63 x+1$
• $y^2=65 x^6+13 x^5+27 x^4+21 x^3+57 x^2+17 x+34$
• $y^2=37 x^6+39 x^5+2 x^4+63 x^3+13 x^2+51 x+23$
• $y^2=54 x^6+71 x^5+31 x^4+75 x^3+46 x^2+29 x+3$
• $y^2=4 x^6+55 x^5+14 x^4+67 x^3+59 x^2+8 x+9$
• $y^2=27 x^6+74 x^5+47 x^4+48 x^3+61 x^2+8 x+50$
• $y^2=2 x^6+64 x^5+62 x^4+65 x^3+25 x^2+24 x+71$
• $y^2=4 x^6+52 x^5+41 x^4+16 x^3+10 x^2+65 x+33$
• $y^2=12 x^6+77 x^5+44 x^4+48 x^3+30 x^2+37 x+20$
• $y^2=70 x^6+x^5+10 x^4+17 x^3+23 x^2+30 x+19$
• $y^2=52 x^6+3 x^5+30 x^4+51 x^3+69 x^2+11 x+57$
• and

• $y^2=12 x^6+34 x^5+12 x^4+67 x^3+23 x^2+46 x+11$
• $y^2=36 x^6+23 x^5+36 x^4+43 x^3+69 x^2+59 x+33$
• $y^2=52 x^6+x^5+26 x^4+65 x^3+43 x^2+29 x+48$
• $y^2=77 x^6+3 x^5+78 x^4+37 x^3+50 x^2+8 x+65$
• $y^2=53 x^6+39 x^5+54 x^4+50 x^3+54 x^2+3 x+77$
• $y^2=x^6+38 x^5+4 x^4+71 x^3+4 x^2+9 x+73$
• $y^2=20 x^6+47 x^5+59 x^4+48 x^2+37 x+7$
• $y^2=60 x^6+62 x^5+19 x^4+65 x^2+32 x+21$
• $y^2=72 x^6+13 x^5+55 x^4+28 x^3+63 x^2+53 x+63$
• $y^2=58 x^6+39 x^5+7 x^4+5 x^3+31 x^2+x+31$
• $y^2=60 x^6+46 x^5+61 x^4+31 x^3+15 x^2+22 x+14$
• $y^2=22 x^6+59 x^5+25 x^4+14 x^3+45 x^2+66 x+42$
• $y^2=17 x^6+30 x^5+49 x^4+68 x^3+40 x^2+39 x+54$
• $y^2=51 x^6+11 x^5+68 x^4+46 x^3+41 x^2+38 x+4$
• $y^2=61 x^6+47 x^5+74 x^4+33 x^3+2 x^2+28 x+66$
• $y^2=25 x^6+62 x^5+64 x^4+20 x^3+6 x^2+5 x+40$
• $y^2=21 x^6+53 x^5+46 x^4+54 x^3+13 x^2+66 x+35$
• $y^2=63 x^6+x^5+59 x^4+4 x^3+39 x^2+40 x+26$
• $y^2=63 x^6+8 x^5+3 x^4+48 x^3+42 x^2+46 x+10$
• $y^2=31 x^6+24 x^5+9 x^4+65 x^3+47 x^2+59 x+30$
• $y^2=8 x^6+78 x^5+28 x^4+46 x^3+56 x^2+60 x+27$
• $y^2=24 x^6+76 x^5+5 x^4+59 x^3+10 x^2+22 x+2$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{79^{2}}$.

Endomorphism algebra over $\F_{79}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{-149})\).

Endomorphism algebra over $\overline{\F}_{79}$

The base change of $A$ to $\F_{79^{2}}$ is 1.6241.afk^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-149}) \)$)$

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
2.79.a_fk	$4$	(not in LMFDB)
2.79.ag_s	$8$	(not in LMFDB)
2.79.g_s	$8$	(not in LMFDB)