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

• Label: 2.79.h_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:	$( 1 + 79 x^{2} )( 1 + 7 x + 79 x^{2} )$
	$1 + 7 x + 158 x^{2} + 553 x^{3} + 6241 x^{4}$
Frobenius angles:	$\pm0.5$, $\pm0.628833084391$
Angle rank:	$1$ (numerical)
Jacobians:	$80$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6960$	$40646400$	$242439600960$	$1516646114380800$	$9468583046696254800$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$87$	$6509$	$491724$	$38938201$	$3077156157$	$243087695822$	$19203906325803$	$1517108809545361$	$119851595650013796$	$9468276084983613749$

Jacobians and polarizations

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

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

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

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.h 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.h : \(\Q(\sqrt{-267}) \).

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

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

• 1.6241.ef : \(\Q(\sqrt{-267}) \).
• 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.ah_gc	$2$	(not in LMFDB)