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Properties

Label 2.19.a_r
Base field $\F_{19}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{19}$
Dimension:  $2$
L-polynomial:  $1 + 17 x^{2} + 361 x^{4}$
Frobenius angles:  $\pm0.323819359775$, $\pm0.676180640225$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{21}, \sqrt{-55})\)
Galois group:  $C_2^2$
Jacobians:  $28$
Cyclic group of points:    yes

This isogeny class is simple but not geometrically 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $379$ $143641$ $47032384$ $17096870025$ $6131069886979$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $20$ $396$ $6860$ $131188$ $2476100$ $47018886$ $893871740$ $16983709348$ $322687697780$ $6131073516156$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):

  • $y^2=17 x^6+13 x^5+12 x^4+9 x^3+12 x^2+9 x+16$
  • $y^2=15 x^6+7 x^5+5 x^4+18 x^3+5 x^2+18 x+13$
  • $y^2=9 x^6+5 x^5+11 x^4+16 x^3+5 x+18$
  • $y^2=18 x^6+10 x^5+3 x^4+13 x^3+10 x+17$
  • $y^2=10 x^6+9 x^5+11 x^3+8 x^2+12 x+4$
  • $y^2=x^6+18 x^5+3 x^3+16 x^2+5 x+8$
  • $y^2=16 x^6+6 x^5+13 x^4+18 x^3+2 x^2+14 x+14$
  • $y^2=13 x^6+12 x^5+7 x^4+17 x^3+4 x^2+9 x+9$
  • $y^2=7 x^6+13 x^5+2 x^3+10 x^2+13 x+13$
  • $y^2=14 x^6+7 x^5+4 x^3+x^2+7 x+7$
  • $y^2=2 x^6+8 x^5+13 x^4+5 x^3+4 x^2+12 x+17$
  • $y^2=7 x^6+6 x^5+12 x^4+5 x^3+5 x+18$
  • $y^2=14 x^6+12 x^5+5 x^4+10 x^3+10 x+17$
  • $y^2=6 x^6+16 x^5+5 x^4+5 x^3+10 x^2+11 x+3$
  • $y^2=12 x^6+13 x^5+10 x^4+10 x^3+x^2+3 x+6$
  • $y^2=11 x^6+9 x^5+12 x^4+2 x^3+14 x^2+11 x+8$
  • $y^2=3 x^6+18 x^5+5 x^4+4 x^3+9 x^2+3 x+16$
  • $y^2=17 x^6+14 x^5+11 x^4+9 x^3+9 x^2+2 x+6$
  • $y^2=15 x^6+9 x^5+3 x^4+18 x^3+18 x^2+4 x+12$
  • $y^2=15 x^6+16 x^5+3 x^4+2 x^3+2 x^2+16 x+6$
  • $y^2=11 x^6+13 x^5+6 x^4+4 x^3+4 x^2+13 x+12$
  • $y^2=11 x^6+x^5+11 x^4+8 x^3+12 x^2+7 x+8$
  • $y^2=17 x^6+5 x^5+10 x^4+11 x^3+9 x^2+5 x+2$
  • $y^2=6 x^6+9 x^5+16 x^4+3 x^3+2 x^2+12 x+15$
  • $y^2=12 x^6+18 x^5+13 x^4+6 x^3+4 x^2+5 x+11$
  • $y^2=x^6+10 x^5+6 x^4+15 x^2+15 x+18$
  • $y^2=9 x^6+12 x^5+13 x^4+5 x^3+10 x^2+2 x+5$
  • $y^2=18 x^6+5 x^5+7 x^4+10 x^3+x^2+4 x+10$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{19^{2}}$.

Endomorphism algebra over $\F_{19}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{21}, \sqrt{-55})\).
Endomorphism algebra over $\overline{\F}_{19}$
The base change of $A$ to $\F_{19^{2}}$ is 1.361.r 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1155}) \)$)$

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.19.a_ar$4$(not in LMFDB)