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

• Label: 2.79.a_ft
• 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:	$( 1 - 3 x + 79 x^{2} )( 1 + 3 x + 79 x^{2} )$
	$1 + 149 x^{2} + 6241 x^{4}$
Frobenius angles:	$\pm0.446022689903$, $\pm0.553977310097$
Angle rank:	$1$ (numerical)
Jacobians:	$90$
Isomorphism classes:	180
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6391$	$40844881$	$243087973744$	$1516351870571769$	$9468276081859884751$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$80$	$6540$	$493040$	$38930644$	$3077056400$	$243088491966$	$19203908986160$	$1517108776788964$	$119851595982618320$	$9468276081092922300$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 90 curves (of which all are hyperelliptic):

• $y^2=66 x^6+24 x^5+37 x^4+44 x^3+25 x^2+49 x+29$
• $y^2=40 x^6+72 x^5+32 x^4+53 x^3+75 x^2+68 x+8$
• $y^2=37 x^6+54 x^5+39 x^4+25 x^3+74 x^2+28 x+28$
• $y^2=32 x^6+4 x^5+38 x^4+75 x^3+64 x^2+5 x+5$
• $y^2=16 x^6+45 x^5+38 x^4+64 x^3+52 x^2+44 x+13$
• $y^2=48 x^6+56 x^5+35 x^4+34 x^3+77 x^2+53 x+39$
• $y^2=71 x^6+52 x^5+62 x^4+6 x^3+20 x^2+75 x+34$
• $y^2=55 x^6+77 x^5+28 x^4+18 x^3+60 x^2+67 x+23$
• $y^2=77 x^6+51 x^5+16 x^4+44 x^3+73 x^2+26 x+74$
• $y^2=73 x^6+74 x^5+48 x^4+53 x^3+61 x^2+78 x+64$
• $y^2=45 x^6+49 x^5+43 x^4+14 x^3+53 x^2+18 x+20$
• $y^2=56 x^6+68 x^5+50 x^4+42 x^3+x^2+54 x+60$
• $y^2=56 x^6+10 x^5+76 x^4+14 x^3+75 x^2+9 x+4$
• $y^2=60 x^6+65 x^5+9 x^4+7 x^3+74 x^2+16 x+55$
• $y^2=22 x^6+37 x^5+27 x^4+21 x^3+64 x^2+48 x+7$
• $y^2=54 x^6+14 x^5+27 x^4+26 x^3+34 x^2+74 x+66$
• $y^2=4 x^6+42 x^5+2 x^4+78 x^3+23 x^2+64 x+40$
• $y^2=78 x^6+31 x^5+20 x^4+3 x^3+25 x^2+4 x+61$
• $y^2=76 x^6+14 x^5+60 x^4+9 x^3+75 x^2+12 x+25$
• $y^2=11 x^6+37 x^5+74 x^4+50 x^2+66 x+60$
• and

• $y^2=31 x^6+x^5+35 x^4+9 x^3+69 x^2+45 x+14$
• $y^2=14 x^6+3 x^5+26 x^4+27 x^3+49 x^2+56 x+42$
• $y^2=67 x^6+16 x^5+55 x^4+33 x^3+58 x^2+7 x+68$
• $y^2=43 x^6+48 x^5+7 x^4+20 x^3+16 x^2+21 x+46$
• $y^2=40 x^6+17 x^5+25 x^4+29 x^3+65 x^2+17 x+76$
• $y^2=41 x^6+51 x^5+75 x^4+8 x^3+37 x^2+51 x+70$
• $y^2=x^6+55 x^5+58 x^4+26 x^3+45 x^2+66 x+44$
• $y^2=3 x^6+7 x^5+16 x^4+78 x^3+56 x^2+40 x+53$
• $y^2=20 x^6+28 x^5+11 x^4+21 x^3+47 x^2+15 x+11$
• $y^2=60 x^6+5 x^5+33 x^4+63 x^3+62 x^2+45 x+33$
• $y^2=60 x^6+32 x^5+4 x^4+16 x^3+58 x^2+13 x+72$
• $y^2=48 x^6+42 x^5+13 x^4+36 x^3+26 x^2+39 x+54$
• $y^2=65 x^6+47 x^5+39 x^4+29 x^3+78 x^2+38 x+4$
• $y^2=64 x^6+62 x^5+32 x^4+x^3+25 x^2+6 x+51$
• $y^2=34 x^6+28 x^5+17 x^4+3 x^3+75 x^2+18 x+74$
• $y^2=29 x^6+57 x^5+74 x^4+53 x^3+68 x^2+78 x+28$
• $y^2=8 x^6+13 x^5+64 x^4+x^3+46 x^2+76 x+5$
• $y^2=17 x^6+26 x^5+65 x^4+13 x^3+16 x^2+13 x+58$
• $y^2=51 x^6+78 x^5+37 x^4+39 x^3+48 x^2+39 x+16$
• $y^2=26 x^6+62 x^5+44 x^4+39 x^3+65 x^2+51 x+32$
• $y^2=78 x^6+28 x^5+53 x^4+38 x^3+37 x^2+74 x+17$
• $y^2=36 x^6+75 x^5+21 x^4+25 x^3+44 x^2+24 x+44$
• $y^2=29 x^6+67 x^5+63 x^4+75 x^3+53 x^2+72 x+53$
• $y^2=45 x^6+34 x^5+23 x^4+43 x^3+2 x^2+47 x+44$
• $y^2=56 x^6+23 x^5+69 x^4+50 x^3+6 x^2+62 x+53$
• $y^2=30 x^6+24 x^5+61 x^4+43 x^3+26 x^2+53 x+25$
• $y^2=12 x^6+39 x^5+47 x^4+20 x^3+45 x^2+64 x+76$
• $y^2=36 x^6+38 x^5+62 x^4+60 x^3+56 x^2+34 x+70$
• $y^2=74 x^6+21 x^5+7 x^4+6 x^3+10 x^2+24 x+25$
• $y^2=64 x^6+63 x^5+21 x^4+18 x^3+30 x^2+72 x+75$
• $y^2=43 x^6+18 x^5+68 x^4+11 x^3+60 x^2+17 x+13$
• $y^2=50 x^6+54 x^5+46 x^4+33 x^3+22 x^2+51 x+39$
• $y^2=61 x^6+55 x^5+35 x^4+35 x^3+36 x^2+47 x+55$
• $y^2=25 x^6+7 x^5+26 x^4+26 x^3+29 x^2+62 x+7$
• $y^2=4 x^6+59 x^5+12 x^4+16 x^3+19 x^2+53 x+53$
• $y^2=47 x^6+40 x^5+3 x^4+51 x^3+4 x^2+36 x+9$
• $y^2=53 x^6+58 x^5+16 x^4+3 x^3+39 x^2+35 x+19$
• $y^2=x^6+16 x^5+48 x^4+9 x^3+38 x^2+26 x+57$
• $y^2=62 x^6+3 x^5+15 x^4+75 x^3+30 x^2+33 x+6$
• $y^2=28 x^6+9 x^5+45 x^4+67 x^3+11 x^2+20 x+18$
• $y^2=4 x^6+48 x^5+48 x^4+76 x^3+62 x^2+48 x+34$
• $y^2=12 x^6+65 x^5+65 x^4+70 x^3+28 x^2+65 x+23$
• $y^2=51 x^6+x^5+34 x^4+62 x^3+66 x^2+19 x+14$
• $y^2=74 x^6+3 x^5+23 x^4+28 x^3+40 x^2+57 x+42$
• $y^2=36 x^6+52 x^5+39 x^4+26 x^3+3 x^2+11 x+41$
• $y^2=29 x^6+77 x^5+38 x^4+78 x^3+9 x^2+33 x+44$
• $y^2=8 x^6+46 x^5+11 x^4+76 x^3+49 x^2+12 x+67$
• $y^2=24 x^6+59 x^5+33 x^4+70 x^3+68 x^2+36 x+43$
• $y^2=52 x^6+4 x^5+4 x^4+36 x^3+12 x^2+54 x+18$
• $y^2=77 x^6+12 x^5+12 x^4+29 x^3+36 x^2+4 x+54$
• $y^2=70 x^6+22 x^5+41 x^4+35 x^3+41 x^2+22 x+70$
• $y^2=52 x^6+66 x^5+44 x^4+26 x^3+44 x^2+66 x+52$
• $y^2=13 x^6+63 x^5+62 x^4+54 x^3+64 x^2+37 x+40$
• $y^2=39 x^6+31 x^5+28 x^4+4 x^3+34 x^2+32 x+41$
• $y^2=3 x^6+38 x^5+42 x^4+66 x^3+76 x^2+33 x+32$
• $y^2=9 x^6+35 x^5+47 x^4+40 x^3+70 x^2+20 x+17$
• $y^2=11 x^6+23 x^5+33 x^4+47 x^3+68 x^2+40 x+74$
• $y^2=33 x^6+69 x^5+20 x^4+62 x^3+46 x^2+41 x+64$
• $y^2=53 x^6+2 x^5+7 x^4+18 x^3+34 x^2+66 x+65$
• $y^2=x^6+6 x^5+21 x^4+54 x^3+23 x^2+40 x+37$
• $y^2=46 x^6+56 x^5+x^4+37 x^3+16 x^2+37 x+1$
• $y^2=59 x^6+10 x^5+3 x^4+32 x^3+48 x^2+32 x+3$
• $y^2=13 x^6+71 x^5+61 x^4+10 x^3+63 x^2+60 x+45$
• $y^2=39 x^6+55 x^5+25 x^4+30 x^3+31 x^2+22 x+56$
• $y^2=30 x^6+78 x^5+2 x^4+60 x^3+62 x^2+18 x+1$
• $y^2=11 x^6+76 x^5+6 x^4+22 x^3+28 x^2+54 x+3$
• $y^2=6 x^6+26 x^5+29 x^4+5 x^3+58 x^2+21 x+27$
• $y^2=18 x^6+78 x^5+8 x^4+15 x^3+16 x^2+63 x+2$
• $y^2=38 x^6+22 x^5+46 x^4+48 x^3+2 x^2+67 x+77$
• $y^2=35 x^6+66 x^5+59 x^4+65 x^3+6 x^2+43 x+73$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{79^{2}}$.

Endomorphism algebra over $\F_{79}$

The isogeny class factors as 1.79.ad $\times$ 1.79.d 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.ad : \(\Q(\sqrt{-307}) \).
• 1.79.d : \(\Q(\sqrt{-307}) \).

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

The base change of $A$ to $\F_{79^{2}}$ is 1.6241.ft^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-307}) \)$)$

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.ag_gl	$2$	(not in LMFDB)
2.79.g_gl	$2$	(not in LMFDB)
2.79.a_aft	$4$	(not in LMFDB)