Abelian variety isogeny class 1.79.ak over $\F_{79}$

• Label: 1.79.ak
• Base field: $\F_{79}$
• Dimension: $1$
• $p$-rank: $1$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{79}$
Dimension:	$1$
L-polynomial:	$1 - 10 x + 79 x^{2}$
Frobenius angles:	$\pm0.309822710654$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-6}) \)
Galois group:	$C_2$
Jacobians:	$8$
Isomorphism classes:	8

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$70$	$6300$	$494410$	$38959200$	$3077039350$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$70$	$6300$	$494410$	$38959200$	$3077039350$	$243086564700$	$19203901424890$	$1517108804668800$	$119851596527581030$	$9468276088490257500$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which 0 are hyperelliptic):

• $y^2=x^3+22 x+22$
• $y^2=x^3+78 x+76$
• $y^2=x^3+42 x+47$
• $y^2=x^3+63 x+31$
• $y^2=x^3+29 x+29$
• $y^2=x^3+10 x+30$
• $y^2=x^3+65 x+65$
• $y^2=x^3+49 x+68$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{79}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-6}) \).

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
1.79.k	$2$	(not in LMFDB)