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

• Label: 2.79.abb_mk
• 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:	$1 - 27 x + 322 x^{2} - 2133 x^{3} + 6241 x^{4}$
Frobenius angles:	$\pm0.00707186922976$, $\pm0.326261464104$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{73})\)
Galois group:	$C_2^2$
Jacobians:	$4$
Isomorphism classes:	4

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4404$	$38420496$	$243086478192$	$1516904038011456$	$9467799959208916524$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$53$	$6157$	$493040$	$38944825$	$3076901663$	$243085500862$	$19203894822473$	$1517108759642449$	$119851595982618320$	$9468276080799780277$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=50x^6+69x^5+2x^4+45x^3+9x^2+17x+54$
• $y^2=x^6+3x^3+56$
• $y^2=3x^6+3x^3+65$
• $y^2=72x^6+5x^5+37x^4+26x^3+9x^2+9x+34$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{79^{6}}$.

Endomorphism algebra over $\F_{79}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{73})\).

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

The base change of $A$ to $\F_{79^{6}}$ is 1.243087455521.acdptq^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-219}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{79^{2}}$

  The base change of $A$ to $\F_{79^{2}}$ is the simple isogeny class 2.6241.adh_blw and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{73})\).

• Endomorphism algebra over $\F_{79^{3}}$

  The base change of $A$ to $\F_{79^{3}}$ is the simple isogeny class 2.493039.a_acdptq and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{73})\).

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.a_dh	$3$	(not in LMFDB)
2.79.bb_mk	$3$	(not in LMFDB)
2.79.a_dh	$6$	(not in LMFDB)