Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 9 x + 42 x^{2} - 171 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.0659896528252$, $\pm0.482872009315$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.257725.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 16 |
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: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $224$ | $130816$ | $46315136$ | $16828693504$ | $6125033342624$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $11$ | $365$ | $6752$ | $129129$ | $2473661$ | $47054486$ | $893886599$ | $16983349009$ | $322687499168$ | $6131072894525$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=18x^6+7x^5+13x^3+3x^2+18x+12$
- $y^2=13x^6+x^5+5x^4+13x^3+10x^2+16x+9$
- $y^2=18x^6+15x^5+13x^4+13x^3+4x^2+13x+2$
- $y^2=13x^6+4x^5+x^4+8x^3+17x^2+x+13$
- $y^2=14x^6+11x^5+10x^4+6x^3+18x$
- $y^2=13x^6+15x^5+6x^4+15x^3+7x^2+18x+2$
- $y^2=18x^6+6x^5+4x^4+3x^3+5x^2+2x+1$
- $y^2=16x^5+x^4+2x^3+9x^2+12x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19}$.
Endomorphism algebra over $\F_{19}$The endomorphism algebra of this simple isogeny class is 4.0.257725.1. |
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 |
---|---|---|
2.19.j_bq | $2$ | (not in LMFDB) |