Properties

Label 3.13.a_f_ack
Base field $\F_{13}$
Dimension $3$
$p$-rank $3$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{13}$
Dimension:  $3$
L-polynomial:  $1 + 5 x^{2} - 62 x^{3} + 65 x^{4} + 2197 x^{6}$
Frobenius angles:  $\pm0.133228324985$, $\pm0.593752241564$, $\pm0.714244339329$
Angle rank:  $3$ (numerical)
Number field:  6.0.284839616.1
Galois group:  $S_4\times C_2$
Cyclic group of points:    yes

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:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $2206$ $5139980$ $9743683606$ $23470896273200$ $51400079040383566$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $14$ $180$ $2012$ $28772$ $372844$ $4826760$ $62765878$ $815791452$ $10604717366$ $137858806400$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 25 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

  • $y^2=x^8+11 x^7+11 x^6+4 x^5+7 x^4+3 x^3+6 x^2+x+1$
  • $y^2=2 x^8+10 x^7+9 x^6+6 x^5+6 x^4+12 x^3+10 x^2+12 x+7$
  • $y^2=x^8+7 x^7+8 x^6+5 x^5+8 x^3+8 x+12$
  • $y^2=x^8+x^7+2 x^6+3 x^4+x^3+9 x^2+8 x+7$
  • $y^2=2 x^8+2 x^7+x^5+x^4+12 x^2+5 x+11$
  • $y^2=x^8+7 x^7+5 x^6+10 x^5+3 x^4+4 x^3+7 x^2+8 x+6$
  • $y^2=2 x^8+7 x^7+8 x^6+6 x^5+x^4+x^3+4 x^2+5 x+12$
  • $y^2=2 x^8+11 x^7+8 x^6+8 x^5+8 x^4+9 x^3+4 x^2+9 x+12$
  • $y^2=2 x^8+8 x^7+12 x^6+12 x^5+6 x^4+x^3+3 x^2+7 x+12$
  • $y^2=2 x^8+3 x^7+9 x^6+6 x^5+x^4+8 x^3+8 x^2+12 x+4$
  • $y^2=x^8+7 x^7+6 x^6+3 x^5+6 x^4+2 x^3+3 x^2+x+12$
  • $y^2=2 x^8+4 x^7+x^6+6 x^5+10 x^4+6 x^3+11 x^2+2 x+1$
  • $y^2=x^8+12 x^6+6 x^5+x^4+4 x^3+4 x^2+8 x+7$
  • $y^2=2 x^8+12 x^7+x^6+3 x^5+11 x^4+10 x^3+2 x^2+6$
  • $y^2=x^8+9 x^7+12 x^6+9 x^5+6 x^4+5 x^3+12 x^2+2$
  • $y^2=2 x^8+11 x^7+2 x^6+7 x^5+4 x^4+4 x^3+6 x^2+7 x+5$
  • $y^2=2 x^8+4 x^7+3 x^6+5 x^4+x^3+4 x^2+12 x+5$
  • $y^2=2 x^8+x^7+11 x^6+9 x^5+2 x^4+12 x^3+9 x^2+2 x+8$
  • $y^2=2 x^8+10 x^7+10 x^6+6 x^5+10 x^4+8 x^3+4 x^2+3 x+11$
  • $y^2=x^8+8 x^7+10 x^5+3 x^4+6 x^3+9 x^2+4 x+10$
  • $y^2=x^8+6 x^7+5 x^6+6 x^5+3 x^4+10 x^3+7 x^2+2 x+10$
  • $y^2=x^8+4 x^7+6 x^6+11 x^5+11 x^4+11 x^3+5 x^2+x+7$
  • $y^2=x^8+2 x^6+10 x^5+10 x^4+4 x^3+3 x^2+3 x+8$
  • $y^2=2 x^8+2 x^6+6 x^5+5 x^4+x^3+3 x^2+3$
  • $y^2=x^8+2 x^7+10 x^6+x^5+7 x^4+12 x^3+12 x^2+12 x+12$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13}$.

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 6.0.284839616.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
3.13.a_f_ck$2$(not in LMFDB)