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

• Label: 2.79.f_gc
• Base field: $\F_{79}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• 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:	$( 1 + 79 x^{2} )( 1 + 5 x + 79 x^{2} )$
	$1 + 5 x + 158 x^{2} + 395 x^{3} + 6241 x^{4}$
Frobenius angles:	$\pm0.5$, $\pm0.590756304637$
Angle rank:	$1$ (numerical)
Jacobians:	$80$

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6800$	$40800000$	$242565819200$	$1516419964800000$	$9468613872647270000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$85$	$6533$	$491980$	$38932393$	$3077166175$	$243088304078$	$19203901001545$	$1517108782793713$	$119851596359466340$	$9468276082884522173$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 80 curves (of which all are hyperelliptic):

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

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

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.a $\times$ 1.79.f 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.a : \(\Q(\sqrt{-79}) \).
• 1.79.f : \(\Q(\sqrt{-291}) \).

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

The base change of $A$ to $\F_{79^{2}}$ is 1.6241.fd $\times$ 1.6241.gc. The endomorphism algebra for each factor is:

• 1.6241.fd : \(\Q(\sqrt{-291}) \).
• 1.6241.gc : the quaternion algebra over \(\Q\) ramified at $79$ and $\infty$.

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.af_gc	$2$	(not in LMFDB)