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

• Label: 2.79.a_afe
• 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:	$1 - 134 x^{2} + 6241 x^{4}$
Frobenius angles:	$\pm0.0888729187302$, $\pm0.911127081270$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-6}, \sqrt{73})\)
Galois group:	$C_2^2$
Jacobians:	$44$
Isomorphism classes:	80
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6108$	$37307664$	$243087558300$	$1516682492273664$	$9468276088408764828$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$80$	$5974$	$493040$	$38939134$	$3077056400$	$243087661078$	$19203908986160$	$1517108905777534$	$119851595982618320$	$9468276094190682454$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 44 curves (of which all are hyperelliptic):

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

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

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 \(\Q(\sqrt{-6}, \sqrt{73})\).

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

The base change of $A$ to $\F_{79^{2}}$ is 1.6241.afe^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-438}) \)$)$

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_fe	$4$	(not in LMFDB)