Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 10 x + 52 x^{2} - 190 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.0969409553786$, $\pm0.438146995596$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.38720.3 |
Galois group: | $D_{4}$ |
Jacobians: | $10$ |
Isomorphism classes: | 10 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $214$ | $131396$ | $46981774$ | $16884386000$ | $6124262327854$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $366$ | $6850$ | $129558$ | $2473350$ | $47054766$ | $893962030$ | $16983757278$ | $322687625770$ | $6131069385406$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=16x^6+x^5+x^4+5x^3+6x^2+x+13$
- $y^2=3x^6+12x^5+18x^4+10x^3+16x^2+13x+16$
- $y^2=18x^6+7x^5+16x^4+x^3+9x^2+4x+1$
- $y^2=x^6+14x^5+16x^4+8x^3+2x^2+12x+18$
- $y^2=7x^6+15x^5+15x^4+8x^3+11x^2+3x+14$
- $y^2=3x^6+x^5+14x^4+8x^3+17x^2+12x+16$
- $y^2=12x^6+15x^5+13x^4+11x^3+7x^2+4x+8$
- $y^2=13x^6+14x^5+18x^4+13x^3+12x^2+4x+12$
- $y^2=8x^6+15x^5+16x^4+18x^3+x^2+5x+7$
- $y^2=13x^6+x^5+9x^4+12x^2+4x+10$
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 4.0.38720.3. |
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.k_ca | $2$ | (not in LMFDB) |