Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 11 x + 61 x^{2} - 209 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.111054296713$, $\pm0.395633834800$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.508805.3 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $203$ | $130529$ | $47425469$ | $16943316845$ | $6124641547248$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $9$ | $363$ | $6915$ | $130011$ | $2473504$ | $47047431$ | $893962281$ | $16984034211$ | $322688618535$ | $6131066008678$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+17x^5+11x^4+x^3+10x^2+11x+14$
- $y^2=14x^6+15x^5+17x^4+x^3+16x^2+12x+14$
- $y^2=15x^5+10x^4+12x^3+3x^2+9x+3$
- $y^2=8x^6+3x^5+17x^4+9x^3+x^2+7x+14$
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 4.0.508805.3. |
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.l_cj | $2$ | (not in LMFDB) |