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

• Label: 2.79.e_acl
• Base field: $\F_{79}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• 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:	$( 1 - 13 x + 79 x^{2} )( 1 + 17 x + 79 x^{2} )$
	$1 + 4 x - 63 x^{2} + 316 x^{3} + 6241 x^{4}$
Frobenius angles:	$\pm0.238910621905$, $\pm0.905577288571$
Angle rank:	$1$ (numerical)
Jacobians:	$50$
Cyclic group of points:	yes

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6499$	$38077641$	$243960917776$	$1517408044499529$	$9468585530762718499$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$6100$	$494808$	$38957764$	$3077156964$	$243087864766$	$19203900223116$	$1517108791019524$	$119851594749153672$	$9468276086585852500$

Jacobians and polarizations

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

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{79^{3}}$.

Endomorphism algebra over $\F_{79}$

The isogeny class factors as 1.79.an $\times$ 1.79.r 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.an : \(\Q(\sqrt{-3}) \).
• 1.79.r : \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{79^{3}}$ is 1.493039.bia^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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.abe_op	$2$	(not in LMFDB)
2.79.ae_acl	$2$	(not in LMFDB)
2.79.be_op	$2$	(not in LMFDB)
2.79.aba_mp	$3$	(not in LMFDB)
2.79.ar_ic	$3$	(not in LMFDB)
2.79.ai_gs	$3$	(not in LMFDB)
2.79.n_dm	$3$	(not in LMFDB)
2.79.bi_rf	$3$	(not in LMFDB)