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

• Label: 2.79.ae_gg
• 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:	$( 1 - 2 x + 79 x^{2} )^{2}$
	$1 - 4 x + 162 x^{2} - 316 x^{3} + 6241 x^{4}$
Frobenius angles:	$\pm0.464111352053$, $\pm0.464111352053$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6084$	$40908816$	$243548172036$	$1516233883567104$	$9467911264180305924$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$76$	$6550$	$493972$	$38927614$	$3076937836$	$243088993366$	$19203921428404$	$1517108713301374$	$119851594806470668$	$9468276087906361750$

Jacobians and polarizations

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

• $y^2=20 x^6+18 x^5+60 x^4+50 x^3+37 x^2+63 x+19$
• $y^2=59 x^6+77 x^5+33 x^4+69 x^3+39 x^2+27 x+39$
• $y^2=x^6+42 x^3+65$
• $y^2=54 x^6+9 x^5+18 x^4+34 x^3+40 x^2+72 x+56$
• $y^2=42 x^6+47 x^5+40 x^4+42 x^3+14 x^2+40 x+46$
• $y^2=19 x^6+16 x^5+53 x^4+39 x^3+46 x^2+3 x+61$
• $y^2=72 x^6+27 x^5+33 x^4+23 x^3+5 x^2+38 x+21$
• $y^2=77 x^6+64 x^5+58 x^4+39 x^3+57 x^2+39 x+4$
• $y^2=35 x^6+22 x^5+74 x^4+3 x^3+53 x^2+20 x+33$
• $y^2=70 x^6+29 x^5+55 x^4+25 x^3+55 x^2+29 x+70$
• $y^2=19 x^6+7 x^5+63 x^4+10 x^3+28 x^2+56 x+50$
• $y^2=58 x^6+68 x^5+44 x^4+71 x^3+67 x^2+15 x+36$
• $y^2=30 x^6+26 x^5+77 x^4+26 x^3+60 x^2+57 x+36$
• $y^2=18 x^6+59 x^5+12 x^4+7 x^3+57 x^2+61 x+53$
• $y^2=16 x^6+48 x^5+44 x^4+26 x^3+18 x^2+37 x+73$
• $y^2=x^6+x^3+46$
• $y^2=53 x^6+54 x^5+73 x^4+15 x^3+55 x^2+74 x+74$
• $y^2=71 x^6+14 x^5+36 x^4+28 x^3+55 x^2+63 x+27$
• $y^2=61 x^6+3 x^5+24 x^4+50 x^3+3 x^2+66$
• $y^2=2 x^6+67 x^5+28 x^4+61 x^3+7 x+42$
• and

• $y^2=49 x^6+40 x^5+62 x^4+41 x^3+29 x^2+37 x+53$
• $y^2=43 x^6+47 x^5+33 x^4+50 x^3+13 x^2+50 x+43$
• $y^2=16 x^6+66 x^5+14 x^4+34 x^3+14 x^2+66 x+16$
• $y^2=32 x^6+35 x^5+53 x^4+x^3+74 x^2+57 x+17$
• $y^2=62 x^6+52 x^5+74 x^3+13 x^2+38 x+77$
• $y^2=71 x^6+27 x^5+63 x^4+63 x^3+3 x^2+15 x+28$
• $y^2=51 x^6+65 x^5+22 x^4+70 x^3+22 x^2+65 x+51$
• $y^2=48 x^6+70 x^5+11 x^4+49 x^3+11 x^2+70 x+48$
• $y^2=42 x^6+4 x^5+6 x^4+23 x^3+x^2+41 x+53$
• $y^2=49 x^6+23 x^5+61 x^4+74 x^3+53 x^2+8 x+25$
• $y^2=45 x^6+57 x^5+56 x^4+41 x^3+75 x^2+40 x+56$
• $y^2=13 x^6+19 x^5+19 x^4+19 x^3+69 x^2+19 x+54$
• $y^2=3 x^6+20 x^5+74 x^4+5 x^3+5 x^2+77 x+39$
• $y^2=7 x^6+x^5+46 x^4+71 x^3+19 x^2+47 x+12$
• $y^2=75 x^6+26 x^5+34 x^4+68 x^3+26 x^2+78 x+24$
• $y^2=47 x^6+57 x^5+63 x^4+22 x^3+32 x^2+41 x+68$
• $y^2=3 x^6+51 x^5+28 x^4+78 x^3+64 x^2+50 x+43$
• $y^2=18 x^6+56 x^5+49 x^4+7 x^3+11 x^2+56 x+17$
• $y^2=5 x^6+24 x^5+35 x^4+15 x^3+35 x^2+24 x+5$
• $y^2=60 x^6+53 x^5+48 x^4+59 x^3+46 x^2+70 x+54$
• $y^2=67 x^6+25 x^5+61 x^4+6 x^3+66 x^2+47 x+12$
• $y^2=4 x^6+16 x^5+41 x^4+2 x^2+71 x+5$
• $y^2=x^6+13 x^5+54 x^4+72 x^3+30 x^2+19 x+28$
• $y^2=25 x^6+6 x^5+26 x^4+58 x^3+30 x^2+12 x+6$
• $y^2=20 x^6+62 x^5+32 x^4+56 x^3+42 x^2+20 x+73$
• $y^2=64 x^6+23 x^5+25 x^4+59 x^3+x^2+8 x+52$
• $y^2=x^6+33 x^5+26 x^4+15 x^3+26 x^2+33 x+1$
• $y^2=63 x^6+51 x^5+25 x^4+34 x^3+27 x^2+41 x+24$
• $y^2=29 x^6+12 x^5+2 x^4+25 x^2+22 x+25$
• $y^2=12 x^6+68 x^5+23 x^4+28 x^3+70 x^2+58 x+19$
• $y^2=76 x^6+46 x^5+13 x^4+64 x^3+17 x^2+27 x+35$
• $y^2=38 x^6+71 x^4+71 x^2+38$
• $y^2=66 x^6+5 x^5+37 x^4+48 x^2+41 x+43$
• $y^2=56 x^6+44 x^4+44 x^2+56$
• $y^2=68 x^6+40 x^5+26 x^4+25 x^3+19 x^2+46 x+29$
• $y^2=56 x^6+36 x^5+16 x^4+4 x^3+5 x^2+12 x+6$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{79}$

The isogeny class factors as 1.79.ac^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-78}) \)$)$

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_fy	$2$	(not in LMFDB)
2.79.e_gg	$2$	(not in LMFDB)
2.79.c_acx	$3$	(not in LMFDB)