Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 12 x + 72 x^{2} - 228 x^{3} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.176318466621$, $\pm0.323681533379$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\zeta_{8})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $5$ |
| Isomorphism classes: | 5 |
This isogeny class is simple but not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $194$ | $130756$ | $48296882$ | $17097131536$ | $6135495123074$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $362$ | $7040$ | $131190$ | $2477888$ | $47045882$ | $893875928$ | $16983707614$ | $322688485640$ | $6131066257802$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=8x^6+18x^5+10x^4+6x^3+15x^2+17x+12$
- $y^2=2x^6+10x^5+18x^3+7x^2+7x+15$
- $y^2=14x^6+16x^5+5x^4+17x^2+15x+4$
- $y^2=3x^6+6x^5+15x^4+4x^3+4x^2+6x+12$
- $y^2=5x^6+15x^5+12x^4+5x^3+16x^2+9x+10$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19^{4}}$.
Endomorphism algebra over $\F_{19}$| The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{8})\). |
| The base change of $A$ to $\F_{19^{4}}$ is 1.130321.qs 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$ |
- Endomorphism algebra over $\F_{19^{2}}$
The base change of $A$ to $\F_{19^{2}}$ is the simple isogeny class 2.361.a_qs and its endomorphism algebra is \(\Q(\zeta_{8})\).
Base change
This is a primitive isogeny class.