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

• Label: 2.79.f_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:	$( 1 + x + 79 x^{2} )( 1 + 4 x + 79 x^{2} )$
	$1 + 5 x + 162 x^{2} + 395 x^{3} + 6241 x^{4}$
Frobenius angles:	$\pm0.517915787826$, $\pm0.572243955238$
Angle rank:	$2$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6804$	$40851216$	$242536445424$	$1516335861096000$	$9468680340163987404$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$85$	$6541$	$491920$	$38930233$	$3077187775$	$243088590526$	$19203896858665$	$1517108758657873$	$119851596935278000$	$9468276083872636981$

Jacobians and polarizations

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

• $y^2=40 x^6+46 x^5+26 x^4+39 x^3+60 x^2+5 x+55$
• $y^2=35 x^6+37 x^5+40 x^4+15 x^3+43 x^2+66 x+1$
• $y^2=5 x^5+75 x^4+62 x^3+31 x^2+44 x+65$
• $y^2=61 x^6+71 x^5+16 x^4+40 x^3+60 x^2+36 x+29$
• $y^2=30 x^6+52 x^5+63 x^4+47 x^3+37 x^2+11 x+51$
• $y^2=41 x^6+64 x^5+63 x^4+75 x^3+58 x^2+22 x+28$
• $y^2=4 x^6+2 x^5+7 x^4+74 x^3+38 x^2+4 x+46$
• $y^2=77 x^5+14 x^4+9 x^3+6 x^2+28 x+45$
• $y^2=19 x^6+60 x^5+7 x^4+38 x^3+49 x^2+35 x+50$
• $y^2=61 x^6+33 x^5+77 x^4+44 x^3+26 x^2+34 x+67$
• $y^2=39 x^6+68 x^5+28 x^4+19 x^3+17 x^2+57 x+53$
• $y^2=74 x^5+67 x^4+71 x^3+62 x^2+73 x+45$
• $y^2=71 x^6+48 x^5+50 x^4+76 x^3+48 x^2+41 x+44$
• $y^2=25 x^6+62 x^5+29 x^4+23 x^3+78 x^2+39 x+46$
• $y^2=45 x^6+2 x^5+36 x^4+53 x^3+62 x^2+40 x+7$
• $y^2=44 x^6+56 x^5+54 x^4+68 x^3+26 x^2+11 x+50$
• $y^2=26 x^6+52 x^5+7 x^4+78 x^3+5 x^2+x+3$
• $y^2=28 x^6+41 x^5+30 x^4+68 x^3+61 x^2+65 x+55$
• $y^2=52 x^6+46 x^5+65 x^4+20 x^3+57 x^2+26 x+71$
• $y^2=20 x^6+9 x^5+18 x^4+52 x^3+18 x^2+43 x+19$
• and

• $y^2=22 x^6+10 x^5+17 x^4+4 x^3+43 x^2+76 x+37$
• $y^2=24 x^6+12 x^5+37 x^4+60 x^3+54 x^2+11 x+42$
• $y^2=45 x^6+71 x^5+46 x^4+18 x^3+9 x^2+52 x+34$
• $y^2=64 x^6+71 x^5+35 x^4+x^3+78 x^2+26 x+47$
• $y^2=24 x^5+23 x^4+13 x^3+7 x^2+68 x+5$
• $y^2=72 x^6+77 x^5+2 x^4+26 x^3+45 x^2+55 x+16$
• $y^2=30 x^6+46 x^5+2 x^3+38 x^2+3 x+7$
• $y^2=54 x^6+21 x^5+76 x^4+38 x^3+72 x^2+45 x+43$
• $y^2=17 x^6+68 x^5+49 x^4+16 x^3+69 x^2+35 x+58$
• $y^2=50 x^6+31 x^5+8 x^4+65 x^3+47 x^2+44 x+57$
• $y^2=62 x^6+30 x^5+58 x^4+29 x^3+65 x^2+51 x+76$
• $y^2=3 x^6+9 x^5+3 x^4+2 x^3+31 x^2+50 x+43$
• $y^2=75 x^6+30 x^5+41 x^4+66 x^3+61 x^2+78 x+37$
• $y^2=55 x^6+61 x^5+18 x^4+22 x^3+8 x^2+78 x+46$
• $y^2=73 x^6+64 x^5+14 x^4+38 x^3+x^2+49 x+52$
• $y^2=60 x^6+41 x^5+18 x^4+67 x^3+35 x^2+8 x+15$
• $y^2=2 x^6+62 x^5+51 x^4+6 x^3+48 x^2+32 x+59$
• $y^2=27 x^6+68 x^5+46 x^4+72 x^3+28 x^2+65 x+29$
• $y^2=40 x^6+24 x^5+61 x^4+14 x^3+13 x^2+7 x+50$
• $y^2=22 x^6+26 x^5+50 x^4+54 x^3+9 x^2+39 x+14$
• $y^2=69 x^5+47 x^4+42 x^3+31 x^2+49 x+64$
• $y^2=44 x^6+72 x^5+4 x^4+60 x^3+67 x^2+6 x+61$
• $y^2=59 x^6+56 x^5+39 x^4+19 x^3+20 x^2+71 x+50$
• $y^2=5 x^6+8 x^5+22 x^4+6 x^3+4 x^2+74 x+74$
• $y^2=14 x^6+51 x^5+60 x^4+72 x^3+77 x^2+43 x+32$
• $y^2=27 x^6+46 x^5+77 x^4+7 x^3+41 x^2+51 x+44$
• $y^2=72 x^6+49 x^5+37 x^4+66 x^3+56 x^2+71 x+22$
• $y^2=5 x^6+50 x^5+4 x^4+10 x^3+33 x^2+64 x+72$
• $y^2=50 x^6+44 x^5+76 x^4+53 x^3+37 x^2+59 x+44$
• $y^2=10 x^6+12 x^5+64 x^4+26 x^3+74 x^2+69$
• $y^2=28 x^6+39 x^5+9 x^4+35 x^3+60 x^2+56 x+76$
• $y^2=19 x^6+77 x^5+63 x^4+72 x^3+52 x^2+36 x+30$
• $y^2=54 x^6+75 x^5+54 x^4+11 x^3+48 x^2+43 x+11$
• $y^2=49 x^6+36 x^5+14 x^4+21 x^3+14 x^2+29 x+21$
• $y^2=53 x^6+47 x^5+10 x^4+54 x^3+63 x^2+40 x+42$
• $y^2=23 x^6+12 x^5+36 x^4+44 x^3+70 x^2+60 x+48$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{79}$

The isogeny class factors as 1.79.b $\times$ 1.79.e and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.79.b : \(\Q(\sqrt{-35}) \).
• 1.79.e : \(\Q(\sqrt{-3}) \).

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.af_gg	$2$	(not in LMFDB)
2.79.ad_fy	$2$	(not in LMFDB)
2.79.d_fy	$2$	(not in LMFDB)
2.79.aq_fl	$3$	(not in LMFDB)
2.79.o_gp	$3$	(not in LMFDB)