Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 11 x + 67 x^{2} - 209 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.225619498788$, $\pm0.332359937866$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.45725.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $209$ | $135641$ | $48809651$ | $17121827789$ | $6134444175024$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $9$ | $375$ | $7113$ | $131379$ | $2477464$ | $47038251$ | $893825991$ | $16983474579$ | $322687702587$ | $6131065430950$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=12x^6+11x^5+13x^4+x^3+4x^2+5x+10$
- $y^2=15x^6+11x^5+13x^4+16x^3+12x^2+16x+2$
- $y^2=10x^6+6x^5+4x^4+6x^3+16x^2+16x$
- $y^2=18x^6+2x^5+3x^4+13x^3+14x^2+8x+7$
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 4.0.45725.1. |
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.l_cp | $2$ | (not in LMFDB) |