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

• Label: 2.47.a_acx
• 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:	$( 1 - 13 x + 47 x^{2} )( 1 + 13 x + 47 x^{2} )$
	$1 - 75 x^{2} + 2209 x^{4}$
Frobenius angles:	$\pm0.102979434792$, $\pm0.897020565208$
Angle rank:	$1$ (numerical)
Jacobians:	$23$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2135$	$4558225$	$10779290480$	$23799518325625$	$52599132692512175$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$48$	$2060$	$103824$	$4877268$	$229345008$	$10779365630$	$506623120464$	$23811303266788$	$1119130473102768$	$52599133149194300$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 23 curves (of which all are hyperelliptic):

• $y^2=14 x^6+3 x^5+x^4+25 x^3+33 x^2+17 x+37$
• $y^2=23 x^6+15 x^5+5 x^4+31 x^3+24 x^2+38 x+44$
• $y^2=27 x^6+11 x^5+18 x^4+27 x^3+30 x^2+41 x+31$
• $y^2=43 x^6+32 x^5+44 x^4+37 x^3+7 x^2+28 x+16$
• $y^2=12 x^6+23 x^5+46 x^4+24 x^3+25 x^2+40 x+30$
• $y^2=36 x^6+13 x^5+43 x^4+3 x^3+28 x^2+26 x+13$
• $y^2=42 x^6+22 x^5+37 x^3+45 x+25$
• $y^2=22 x^6+16 x^5+44 x^3+37 x+31$
• $y^2=24 x^6+4 x^5+7 x^4+30 x^3+39 x^2+34 x+22$
• $y^2=18 x^6+25 x^5+34 x^4+28 x^3+30 x^2+19 x+42$
• $y^2=43 x^6+31 x^5+29 x^4+46 x^3+9 x^2+x+22$
• $y^2=26 x^6+27 x^5+44 x^4+29 x^3+22 x^2+46 x+44$
• $y^2=36 x^6+41 x^5+32 x^4+4 x^3+16 x^2+42 x+32$
• $y^2=33 x^6+23 x^5+19 x^4+43 x^3+37 x^2+11 x+43$
• $y^2=24 x^6+21 x^5+x^4+27 x^3+44 x^2+8 x+27$
• $y^2=31 x^6+11 x^5+34 x^4+29 x^3+7 x^2+27 x+8$
• $y^2=14 x^6+8 x^5+29 x^4+4 x^3+35 x^2+41 x+40$
• $y^2=33 x^6+15 x^5+34 x^4+20 x^3+x^2+2 x+13$
• $y^2=24 x^6+28 x^5+29 x^4+6 x^3+5 x^2+10 x+18$
• $y^2=24 x^6+42 x^5+x^4+5 x^3+10 x^2+17 x+30$
• $y^2=27 x^6+9 x^5+16 x^4+39 x^3+22 x^2+6 x+13$
• $y^2=x^6+25 x^5+13 x^4+42 x^3+41 x^2+14 x+44$
• $y^2=5 x^6+31 x^5+18 x^4+22 x^3+17 x^2+23 x+32$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{47^{2}}$.

Endomorphism algebra over $\F_{47}$

The isogeny class factors as 1.47.an $\times$ 1.47.n 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.n : \(\Q(\sqrt{-19}) \).

Endomorphism algebra over $\overline{\F}_{47}$

The base change of $A$ to $\F_{47^{2}}$ is 1.2209.acx^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$

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
2.47.aba_kd	$2$	(not in LMFDB)
2.47.ba_kd	$2$	(not in LMFDB)
2.47.a_cx	$4$	(not in LMFDB)