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

• Label: 2.79.aj_gw
• 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: no

Invariants

Base field:	$\F_{79}$
Dimension:	$2$
L-polynomial:	$( 1 - 5 x + 79 x^{2} )( 1 - 4 x + 79 x^{2} )$
	$1 - 9 x + 178 x^{2} - 711 x^{3} + 6241 x^{4}$
Frobenius angles:	$\pm0.409243695363$, $\pm0.427756044762$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	40
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5700$	$40698000$	$244047848400$	$1516606900200000$	$9467628874854253500$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$71$	$6517$	$494984$	$38937193$	$3076846061$	$243087522622$	$19203925733819$	$1517108879580913$	$119851594989037976$	$9468276072771404077$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{79}$

The isogeny class factors as 1.79.af $\times$ 1.79.ae 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.af : \(\Q(\sqrt{-291}) \).
• 1.79.ae : \(\Q(\sqrt{-3}) \).

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.ab_fi	$2$	(not in LMFDB)
2.79.b_fi	$2$	(not in LMFDB)
2.79.j_gw	$2$	(not in LMFDB)
2.79.as_ip	$3$	(not in LMFDB)
2.79.m_cv	$3$	(not in LMFDB)