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

• Label: 2.79.a_afq
• 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:	$1 - 146 x^{2} + 6241 x^{4}$
Frobenius angles:	$\pm0.0624288022791$, $\pm0.937571197721$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{19})\)
Galois group:	$C_2^2$
Jacobians:	$26$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6096$	$37161216$	$243087076944$	$1516420795797504$	$9468276084969200976$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$80$	$5950$	$493040$	$38932414$	$3077056400$	$243086698366$	$19203908986160$	$1517108809627774$	$119851595982618320$	$9468276087311554750$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 26 curves (of which all are hyperelliptic):

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

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 \(\Q(\sqrt{-3}, \sqrt{19})\).

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

The base change of $A$ to $\F_{79^{2}}$ is 1.6241.afq^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-57}) \)$)$

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_dn	$3$	(not in LMFDB)
2.79.g_dn	$3$	(not in LMFDB)
2.79.a_fq	$4$	(not in LMFDB)