Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 9 x + 43 x^{2} - 171 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.0855093505514$, $\pm0.478262641089$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.18605.1 |
Galois group: | $D_{4}$ |
Jacobians: | $15$ |
Isomorphism classes: | 15 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $225$ | $131625$ | $46500075$ | $16848658125$ | $6126704118000$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $11$ | $367$ | $6779$ | $129283$ | $2474336$ | $47059027$ | $893917469$ | $16983482803$ | $322687893881$ | $6131074547302$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 15 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=8x^6+2x^5+14x^4+5x^3+12x^2+2x+8$
- $y^2=16x^6+12x^5+10x^4+5x^3+10x^2+5x+17$
- $y^2=9x^6+12x^5+2x^3+2x^2+3x+12$
- $y^2=18x^6+18x^5+10x^4+6x^3+15x^2+14x+8$
- $y^2=2x^6+11x^5+18x^4+18x^3+7x^2+18x+5$
- $y^2=9x^6+4x^5+8x^4+12x^3+5x^2+12x+1$
- $y^2=12x^6+3x^5+17x^4+3x^3+3x+14$
- $y^2=17x^6+12x^5+16x^4+15x^3+16x^2+13x+4$
- $y^2=2x^6+14x^5+8x^4+9x^3+7x^2+13x+18$
- $y^2=2x^6+10x^5+2x^4+13x^3+12x^2+14x+5$
- $y^2=10x^6+13x^5+14x^4+x^3+14x^2+17x+3$
- $y^2=5x^6+2x^5+4x^4+18x^3+15x^2+11x+10$
- $y^2=14x^6+10x^5+7x^4+10x^3+11x+15$
- $y^2=15x^6+7x^4+5x^3+7x^2+13x+6$
- $y^2=11x^6+8x^5+13x^4+8x^3+10x^2+18x+18$
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 4.0.18605.1. |
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_br | $2$ | (not in LMFDB) |