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

• Label: 2.79.ak_v
• 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:	$1 - 10 x + 21 x^{2} - 790 x^{3} + 6241 x^{4}$
Frobenius angles:	$\pm0.0235106226797$, $\pm0.643156043987$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{2}, \sqrt{-3})\)
Galois group:	$C_2^2$
Jacobians:	$63$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5463$	$38585169$	$241739388900$	$1516753707246729$	$9468223616028873303$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$70$	$6184$	$490300$	$38940964$	$3077039350$	$243085673878$	$19203901424890$	$1517108815144324$	$119851594892692900$	$9468276076763436904$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 63 curves (of which all are hyperelliptic):

• $y^2=14 x^6+13 x^5+29 x^4+55 x^3+71 x^2+61 x+48$
• $y^2=32 x^6+74 x^5+62 x^4+41 x^3+54 x^2+74 x+73$
• $y^2=x^6+3 x^3+26$
• $y^2=61 x^6+5 x^5+26 x^4+35 x^3+36 x^2+26 x+44$
• $y^2=31 x^6+29 x^5+68 x^4+18 x^3+14 x^2+24 x+67$
• $y^2=49 x^6+69 x^5+52 x^4+34 x^3+14 x^2+59 x+71$
• $y^2=46 x^6+39 x^5+78 x^4+42 x^3+54 x^2+8 x+60$
• $y^2=53 x^6+27 x^4+18 x^3+67 x^2+35 x+15$
• $y^2=18 x^6+49 x^5+10 x^4+34 x^3+13 x^2+12 x+45$
• $y^2=49 x^6+4 x^5+38 x^4+60 x^3+23 x^2+75 x+21$
• $y^2=68 x^6+11 x^5+24 x^4+15 x^3+45 x^2+76 x+47$
• $y^2=62 x^6+42 x^5+10 x^4+14 x^3+42 x^2+69 x+36$
• $y^2=63 x^6+27 x^5+71 x^4+7 x^3+28 x^2+9 x+34$
• $y^2=61 x^6+51 x^5+70 x^4+78 x^3+72 x^2+42 x+28$
• $y^2=58 x^6+22 x^5+49 x^4+69 x^3+17 x^2+78 x+53$
• $y^2=6 x^6+61 x^5+42 x^4+11 x^3+69 x^2+27 x+52$
• $y^2=x^6+x^3+5$
• $y^2=78 x^6+38 x^5+27 x^4+43 x^3+67 x^2+4 x+66$
• $y^2=60 x^6+5 x^5+8 x^4+50 x^3+63 x^2+x+70$
• $y^2=59 x^6+54 x^5+66 x^4+25 x^3+44 x^2+68 x+77$
• and

• $y^2=40 x^6+45 x^5+5 x^4+64 x^3+20 x^2+48 x+32$
• $y^2=48 x^6+72 x^5+14 x^4+30 x^3+48 x^2+7 x+59$
• $y^2=62 x^6+20 x^5+60 x^4+33 x^3+65 x^2+58 x+56$
• $y^2=77 x^6+23 x^5+22 x^4+10 x^3+27 x^2+61 x+60$
• $y^2=38 x^6+15 x^5+14 x^4+64 x^3+71 x^2+41 x+60$
• $y^2=x^6+x^3+45$
• $y^2=50 x^6+57 x^5+42 x^4+61 x^3+28 x^2+13 x+23$
• $y^2=47 x^6+2 x^5+72 x^4+70 x^3+72 x^2+56 x+63$
• $y^2=77 x^6+63 x^5+30 x^4+64 x^3+14 x^2+54 x+11$
• $y^2=30 x^6+51 x^5+46 x^3+24 x^2+2 x+15$
• $y^2=x^6+3 x^3+32$
• $y^2=36 x^6+68 x^5+41 x^4+11 x^3+19 x^2+57 x+12$
• $y^2=x^6+x^3+40$
• $y^2=22 x^6+52 x^5+x^4+54 x^3+38 x^2+71 x+19$
• $y^2=4 x^6+21 x^5+50 x^4+33 x^3+9 x^2+19 x+39$
• $y^2=9 x^6+40 x^5+22 x^4+72 x^3+23 x^2+43 x+18$
• $y^2=25 x^6+20 x^5+55 x^4+18 x^3+6 x^2+24 x+43$
• $y^2=18 x^6+34 x^5+28 x^4+69 x^3+45 x^2+66 x+69$
• $y^2=68 x^6+29 x^5+41 x^4+51 x^3+70 x^2+29 x+26$
• $y^2=70 x^6+54 x^5+53 x^4+76 x^3+x^2+36 x+5$
• $y^2=36 x^6+61 x^5+34 x^4+10 x^3+24 x^2+34 x+29$
• $y^2=55 x^6+61 x^5+9 x^4+68 x^3+52 x^2+2 x+68$
• $y^2=51 x^6+57 x^4+56 x^3+69 x^2+42 x+7$
• $y^2=72 x^6+47 x^5+10 x^4+4 x^3+40 x^2+3 x+67$
• $y^2=45 x^6+56 x^5+33 x^4+20 x^3+65 x^2+51 x+66$
• $y^2=34 x^6+65 x^5+21 x^4+31 x^3+62 x^2+75 x+30$
• $y^2=x^6+3 x^3+31$
• $y^2=42 x^6+19 x^5+30 x^4+40 x^3+19 x^2+42 x+1$
• $y^2=62 x^6+67 x^5+73 x^4+58 x^3+7 x^2+22 x+20$
• $y^2=17 x^6+73 x^5+16 x^4+3 x^3+8 x^2+29 x+76$
• $y^2=47 x^6+75 x^5+33 x^4+42 x^3+17 x^2+46 x+38$
• $y^2=8 x^6+44 x^5+69 x^4+19 x^3+11 x^2+30 x+41$
• $y^2=7 x^6+30 x^5+19 x^4+20 x^3+28 x^2+14 x+59$
• $y^2=46 x^6+66 x^5+25 x^4+48 x^3+20 x^2+56 x+32$
• $y^2=16 x^6+27 x^5+29 x^4+76 x^3+48 x^2+10 x+34$
• $y^2=34 x^6+41 x^5+57 x^4+37 x^3+40 x^2+61 x+52$
• $y^2=29 x^6+14 x^5+73 x^4+18 x^3+56 x^2+16 x+54$
• $y^2=21 x^6+13 x^5+14 x^4+x^3+73 x^2+40 x+44$
• $y^2=x^6+x^3+51$
• $y^2=76 x^6+63 x^5+24 x^4+50 x^3+46 x^2+17 x+61$
• $y^2=7 x^6+24 x^5+28 x^4+16 x^3+x^2+54 x+6$
• $y^2=75 x^6+42 x^5+36 x^4+56 x^3+2 x^2+41 x+7$
• $y^2=4 x^6+32 x^5+40 x^4+27 x^3+41 x^2+59 x+39$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{79^{3}}$.

Endomorphism algebra over $\F_{79}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{2}, \sqrt{-3})\).

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

The base change of $A$ to $\F_{79^{3}}$ is 1.493039.acas^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$

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
2.79.k_v	$2$	(not in LMFDB)
2.79.u_jy	$3$	(not in LMFDB)
2.79.au_jy	$6$	(not in LMFDB)