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

• Label: 2.79.e_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:	$\pm0.535888647947$, $\pm0.535888647947$
Angle rank:	$1$ (numerical)
Jacobians:	$56$
Cyclic group of points:	no
Non-cyclic primes:	$2, 41$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6724$	$40908816$	$242629145476$	$1516233883567104$	$9468640920410325124$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$6550$	$492108$	$38927614$	$3077174964$	$243088993366$	$19203896543916$	$1517108713301374$	$119851597158765972$	$9468276087906361750$

Jacobians and polarizations

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

• $y^2=60 x^6+54 x^5+22 x^4+71 x^3+32 x^2+31 x+57$
• $y^2=46 x^6+52 x^5+11 x^4+23 x^3+13 x^2+9 x+13$
• $y^2=3 x^6+47 x^3+37$
• $y^2=4 x^6+27 x^5+54 x^4+23 x^3+41 x^2+58 x+10$
• $y^2=47 x^6+62 x^5+41 x^4+47 x^3+42 x^2+41 x+59$
• $y^2=59 x^6+58 x^5+44 x^4+13 x^3+68 x^2+x+73$
• $y^2=24 x^6+9 x^5+11 x^4+34 x^3+28 x^2+39 x+7$
• $y^2=52 x^6+74 x^5+72 x^4+13 x^3+19 x^2+13 x+54$
• $y^2=26 x^6+66 x^5+64 x^4+9 x^3+x^2+60 x+20$
• $y^2=76 x^6+36 x^5+71 x^4+61 x^3+71 x^2+36 x+76$
• $y^2=57 x^6+21 x^5+31 x^4+30 x^3+5 x^2+10 x+71$
• $y^2=16 x^6+46 x^5+53 x^4+55 x^3+43 x^2+45 x+29$
• $y^2=10 x^6+35 x^5+52 x^4+35 x^3+20 x^2+19 x+12$
• $y^2=6 x^6+46 x^5+4 x^4+55 x^3+19 x^2+73 x+44$
• $y^2=58 x^6+16 x^5+41 x^4+35 x^3+6 x^2+65 x+77$
• $y^2=3 x^6+3 x^3+59$
• $y^2=44 x^6+18 x^5+77 x^4+5 x^3+71 x^2+51 x+51$
• $y^2=55 x^6+42 x^5+29 x^4+5 x^3+7 x^2+31 x+2$
• $y^2=25 x^6+9 x^5+72 x^4+71 x^3+9 x^2+40$
• $y^2=27 x^6+75 x^5+62 x^4+73 x^3+55 x+14$
• and

• $y^2=69 x^6+66 x^5+47 x^4+40 x^3+36 x^2+65 x+44$
• $y^2=50 x^6+62 x^5+20 x^4+71 x^3+39 x^2+71 x+50$
• $y^2=58 x^6+22 x^5+31 x^4+64 x^3+31 x^2+22 x+58$
• $y^2=37 x^6+38 x^5+44 x^4+53 x^3+51 x^2+19 x+32$
• $y^2=28 x^6+77 x^5+64 x^3+39 x^2+35 x+73$
• $y^2=55 x^6+2 x^5+31 x^4+31 x^3+9 x^2+45 x+5$
• $y^2=17 x^6+48 x^5+60 x^4+76 x^3+60 x^2+48 x+17$
• $y^2=16 x^6+76 x^5+30 x^4+69 x^3+30 x^2+76 x+16$
• $y^2=14 x^6+54 x^5+2 x^4+34 x^3+53 x^2+40 x+44$
• $y^2=68 x^6+69 x^5+25 x^4+64 x^3+x^2+24 x+75$
• $y^2=56 x^6+13 x^5+10 x^4+44 x^3+67 x^2+41 x+10$
• $y^2=39 x^6+57 x^5+57 x^4+57 x^3+49 x^2+57 x+4$
• $y^2=x^6+33 x^5+51 x^4+28 x^3+28 x^2+52 x+13$
• $y^2=21 x^6+3 x^5+59 x^4+55 x^3+57 x^2+62 x+36$
• $y^2=25 x^6+35 x^5+64 x^4+49 x^3+35 x^2+26 x+8$
• $y^2=42 x^6+19 x^5+21 x^4+60 x^3+37 x^2+40 x+49$
• $y^2=x^6+17 x^5+62 x^4+26 x^3+74 x^2+43 x+67$
• $y^2=6 x^6+45 x^5+69 x^4+55 x^3+30 x^2+45 x+32$
• $y^2=15 x^6+72 x^5+26 x^4+45 x^3+26 x^2+72 x+15$
• $y^2=22 x^6+x^5+65 x^4+19 x^3+59 x^2+52 x+4$
• $y^2=75 x^6+61 x^5+73 x^4+2 x^3+22 x^2+42 x+4$
• $y^2=54 x^6+58 x^5+40 x^4+27 x^2+50 x+28$
• $y^2=53 x^6+57 x^5+18 x^4+24 x^3+10 x^2+59 x+62$
• $y^2=75 x^6+18 x^5+78 x^4+16 x^3+11 x^2+36 x+18$
• $y^2=60 x^6+28 x^5+17 x^4+10 x^3+47 x^2+60 x+61$
• $y^2=74 x^6+34 x^5+61 x^4+46 x^3+53 x^2+29 x+70$
• $y^2=53 x^6+11 x^5+35 x^4+5 x^3+35 x^2+11 x+53$
• $y^2=21 x^6+17 x^5+61 x^4+64 x^3+9 x^2+40 x+8$
• $y^2=8 x^6+36 x^5+6 x^4+75 x^2+66 x+75$
• $y^2=36 x^6+46 x^5+69 x^4+5 x^3+52 x^2+16 x+57$
• $y^2=70 x^6+59 x^5+39 x^4+34 x^3+51 x^2+2 x+26$
• $y^2=39 x^6+50 x^4+50 x^2+39$
• $y^2=40 x^6+15 x^5+32 x^4+65 x^2+44 x+50$
• $y^2=10 x^6+53 x^4+53 x^2+10$
• $y^2=49 x^6+66 x^5+35 x^4+61 x^3+59 x^2+68 x+36$
• $y^2=45 x^6+12 x^5+58 x^4+54 x^3+28 x^2+4 x+2$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{79}$

The isogeny class factors as 1.79.c^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.ae_gg	$2$	(not in LMFDB)
2.79.a_fy	$2$	(not in LMFDB)
2.79.ac_acx	$3$	(not in LMFDB)