Properties

Label 3.7.ae_v_aca
Base field $\F_{7}$
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:  $\F_{7}$
Dimension:  $3$
L-polynomial:  $1 - 4 x + 21 x^{2} - 52 x^{3} + 147 x^{4} - 196 x^{5} + 343 x^{6}$
Frobenius angles:  $\pm0.252743015585$, $\pm0.427555044385$, $\pm0.554599990253$
Angle rank:  $3$ (numerical)
Number field:  6.0.56337088.1
Galois group:  $S_4\times C_2$
Cyclic group of points:    no
Non-cyclic primes:   $2$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $260$ $198640$ $44383820$ $13618758400$ $4785930605300$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $76$ $376$ $2364$ $16944$ $118108$ $822308$ $5760252$ $40346764$ $282451516$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 9 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

  • $y^2=x^7+x^6+2 x^4+3 x^3+2 x+3$
  • $y^2=x^7+x^6+5 x^4+4 x^3+6 x^2+6$
  • $y^2=x^7+3 x^6+2 x^4+x^3+6 x^2+5 x+1$
  • $y^2=x^8+x^5+x^4+5 x^3+2 x^2+6 x+6$
  • $y^2=x^8+3 x^5+x^4+3 x^3+5 x^2+3 x+1$
  • $y^2=x^8+x^6+3 x^4+x^3+5 x^2+5 x+6$
  • $y^2=x^8+x^6+2 x^5+5 x^4+3 x^3+4 x^2+2 x+1$
  • $y^2=x^8+x^6+3 x^5+3 x^4+2 x^3+2 x^2+2 x+3$
  • $y^2=x^8+x^6+3 x^5+3 x^4+3 x^3+x^2+3 x+2$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7}$.

Endomorphism algebra over $\F_{7}$
The endomorphism algebra of this simple isogeny class is 6.0.56337088.1.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
3.7.e_v_ca$2$(not in LMFDB)