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

• Label: 2.79.ak_hb
• Base field: $\F_{79}$
• 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:	$\F_{79}$
Dimension:	$2$
L-polynomial:	$( 1 - 5 x + 79 x^{2} )^{2}$
	$1 - 10 x + 183 x^{2} - 790 x^{3} + 6241 x^{4}$
Frobenius angles:	$\pm0.409243695363$, $\pm0.409243695363$
Angle rank:	$1$ (numerical)
Jacobians:	$56$
Cyclic group of points:	no
Non-cyclic primes:	$3, 5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5625$	$40640625$	$244134810000$	$1516703288765625$	$9467600533098890625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$70$	$6508$	$495160$	$38939668$	$3076836850$	$243087180478$	$19203924955390$	$1517108911481188$	$119851595228922280$	$9468276070833971548$

Jacobians and polarizations

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

• $y^2=68 x^6+72 x^5+34 x^4+13 x^3+34 x^2+72 x+68$
• $y^2=19 x^6+17 x^5+11 x^4+31 x^3+10 x^2+78 x+2$
• $y^2=54 x^6+50 x^5+38 x^4+66 x^3+13 x^2+63 x+30$
• $y^2=73 x^6+21 x^5+30 x^4+2 x^3+x^2+45 x+25$
• $y^2=48 x^6+10 x^5+7 x^4+51 x^3+78 x^2+40 x+1$
• $y^2=54 x^6+5 x^5+52 x^4+46 x^3+52 x^2+5 x+54$
• $y^2=52 x^6+50 x^5+13 x^4+12 x^3+42 x^2+63 x+13$
• $y^2=50 x^6+34 x^5+77 x^4+46 x^3+77 x^2+34 x+50$
• $y^2=34 x^6+51 x^5+53 x^4+12 x^3+53 x^2+51 x+34$
• $y^2=46 x^6+16 x^5+44 x^4+22 x^3+44 x^2+16 x+46$
• $y^2=30 x^6+77 x^5+57 x^4+60 x^3+75 x^2+68 x+41$
• $y^2=70 x^6+40 x^5+34 x^4+9 x^3+25 x^2+46 x+66$
• $y^2=56 x^6+46 x^5+71 x^4+71 x^3+71 x^2+46 x+56$
• $y^2=59 x^6+66 x^5+24 x^4+30 x^3+71 x^2+31 x+41$
• $y^2=60 x^6+23 x^5+46 x^4+x^3+35 x^2+63 x+4$
• $y^2=72 x^6+28 x^5+70 x^4+36 x^3+44 x^2+77 x+42$
• $y^2=35 x^6+3 x^5+23 x^4+11 x^3+49 x^2+18 x+29$
• $y^2=29 x^6+17 x^5+41 x^4+35 x^3+23 x^2+2 x+3$
• $y^2=38 x^6+65 x^5+16 x^4+22 x^3+16 x^2+65 x+38$
• $y^2=21 x^6+65 x^5+20 x^4+12 x^3+11 x^2+71 x+41$
• and

• $y^2=10 x^6+44 x^5+65 x^4+9 x^3+16 x^2+22 x+38$
• $y^2=59 x^6+66 x^5+78 x^4+62 x^3+33 x^2+52 x+77$
• $y^2=61 x^6+3 x^5+40 x^4+14 x^3+40 x^2+3 x+61$
• $y^2=61 x^6+64 x^5+27 x^4+64 x^3+66 x^2+63 x+36$
• $y^2=63 x^6+49 x^5+74 x^4+77 x^3+74 x^2+49 x+63$
• $y^2=24 x^6+59 x^5+45 x^4+45 x^3+21 x^2+59 x+77$
• $y^2=52 x^6+36 x^5+9 x^4+34 x^3+43 x^2+73 x+51$
• $y^2=22 x^6+48 x^5+32 x^4+73 x^3+7 x^2+22 x+39$
• $y^2=11 x^6+65 x^5+35 x^4+69 x^3+52 x^2+64 x+45$
• $y^2=70 x^6+55 x^5+53 x^4+24 x^3+53 x^2+55 x+70$
• $y^2=19 x^6+55 x^5+74 x^4+46 x^3+74 x^2+55 x+19$
• $y^2=38 x^6+25 x^5+67 x^4+27 x^3+67 x^2+25 x+38$
• $y^2=6 x^6+29 x^5+16 x^4+23 x^3+78 x^2+61 x+58$
• $y^2=x^6+16 x^3+67$
• $y^2=68 x^6+49 x^5+65 x^4+34 x^3+65 x^2+49 x+68$
• $y^2=43 x^6+10 x^5+15 x^4+8 x^3+11 x^2+3 x+12$
• $y^2=20 x^6+29 x^5+8 x^4+49 x^3+8 x^2+29 x+20$
• $y^2=7 x^6+58 x^5+69 x^4+68 x^3+43 x^2+9 x+68$
• $y^2=30 x^6+75 x^5+x^4+41 x^3+x^2+75 x+30$
• $y^2=x^6+9 x^3+18$
• $y^2=51 x^6+26 x^5+29 x^4+76 x^3+64 x^2+19 x+35$
• $y^2=26 x^6+67 x^5+29 x^4+20 x^3+43 x^2+42 x+68$
• $y^2=78 x^6+70 x^5+71 x^4+8 x^3+60 x^2+33 x+15$
• $y^2=30 x^6+40 x^5+62 x^4+48 x^3+36 x^2+8 x+21$
• $y^2=15 x^6+46 x^5+12 x^4+75 x^3+12 x^2+46 x+15$
• $y^2=52 x^6+21 x^5+9 x^4+38 x^3+56 x^2+54 x+77$
• $y^2=75 x^6+35 x^5+21 x^4+9 x^3+69 x^2+54 x+68$
• $y^2=68 x^6+33 x^5+48 x^4+32 x^3+48 x^2+43 x+25$
• $y^2=8 x^6+36 x^5+67 x^4+41 x^3+37 x^2+57 x+42$
• $y^2=59 x^6+42 x^5+11 x^4+7 x^3+78 x^2+14 x+6$
• $y^2=19 x^6+29 x^5+22 x^4+49 x^3+22 x^2+29 x+19$
• $y^2=37 x^6+11 x^5+42 x^4+67 x^3+42 x^2+11 x+37$
• $y^2=34 x^6+60 x^5+48 x^4+67 x^3+34 x^2+8 x+39$
• $y^2=50 x^6+46 x^5+16 x^4+23 x^3+16 x^2+46 x+50$
• $y^2=4 x^6+10 x^5+27 x^4+46 x^3+27 x^2+10 x+4$
• $y^2=75 x^6+69 x^5+9 x^4+52 x^3+16 x^2+7 x+35$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{79}$

The isogeny class factors as 1.79.af^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-291}) \)$)$

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.a_fd	$2$	(not in LMFDB)
2.79.k_hb	$2$	(not in LMFDB)
2.79.f_acc	$3$	(not in LMFDB)