Abelian variety isogeny class 2.47.av_hq over $\F_{47}$

• Label: 2.47.av_hq
• Base field: $\F_{47}$
• 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:	$\F_{47}$
Dimension:	$2$
L-polynomial:	$( 1 - 13 x + 47 x^{2} )( 1 - 8 x + 47 x^{2} )$
	$1 - 21 x + 198 x^{2} - 987 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.102979434792$, $\pm0.301698511018$
Angle rank:	$2$ (numerical)
Jacobians:	$12$
Isomorphism classes:	78
Cyclic group of points:	no
Non-cyclic primes:	$2, 5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1400$	$4782400$	$10805362400$	$23822569120000$	$52599272364485000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$27$	$2165$	$104076$	$4881993$	$229345617$	$10779118670$	$506622694311$	$23811292347313$	$1119130580128932$	$52599133150549325$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

• $y^2=41 x^6+36 x^5+25 x^4+11 x^3+38 x^2+37 x+35$
• $y^2=44 x^6+37 x^5+45 x^4+8 x^3+33 x^2+28 x+43$
• $y^2=26 x^6+39 x^5+13 x^4+32 x^3+3 x^2+13 x+14$
• $y^2=20 x^6+7 x^5+3 x^4+16 x^3+29 x^2+22 x+43$
• $y^2=40 x^6+18 x^5+7 x^4+12 x^3+25 x^2+31 x+20$
• $y^2=22 x^6+45 x^5+39 x^4+26 x^3+11 x+21$
• $y^2=13 x^6+34 x^5+36 x^4+43 x^3+20 x^2+28 x+20$
• $y^2=11 x^6+45 x^5+18 x^4+3 x^3+9 x^2+28 x+35$
• $y^2=40 x^6+39 x^5+12 x^4+43 x^3+28 x^2+36 x+37$
• $y^2=23 x^6+43 x^5+12 x^4+39 x^3+25 x^2+31 x+19$
• $y^2=x^6+39 x^5+37 x^4+28 x^3+37 x^2+14 x+21$
• $y^2=9 x^6+20 x^5+7 x^4+45 x^3+16 x^2+19 x+35$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{47}$.

Endomorphism algebra over $\F_{47}$

The isogeny class factors as 1.47.an $\times$ 1.47.ai and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.47.an : \(\Q(\sqrt{-19}) \).
• 1.47.ai : \(\Q(\sqrt{-31}) \).

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
2.47.af_ak	$2$	(not in LMFDB)
2.47.f_ak	$2$	(not in LMFDB)
2.47.v_hq	$2$	(not in LMFDB)