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

• Label: 2.79.a_aeq
• Base field: $\F_{79}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{79}$
Dimension:	$2$
L-polynomial:	$1 - 120 x^{2} + 6241 x^{4}$
Frobenius angles:	$\pm0.112723385156$, $\pm0.887276614844$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-38}, \sqrt{278})\)
Galois group:	$C_2^2$
Jacobians:	$50$
Cyclic group of points:	yes

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6122$	$37478884$	$243087974282$	$1516959478970896$	$9468276088295838602$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$80$	$6002$	$493040$	$38946246$	$3077056400$	$243088493042$	$19203908986160$	$1517108958349438$	$119851595982618320$	$9468276093964830002$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 50 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{79}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-38}, \sqrt{278})\).

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

The base change of $A$ to $\F_{79^{2}}$ is 1.6241.aeq^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2641}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.79.a_eq	$4$	(not in LMFDB)