Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 12 x + 67 x^{2} - 228 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.0409513241932$, $\pm0.374284657527$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-3}, \sqrt{7})\) |
Galois group: | $C_2^2$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $189$ | $126441$ | $47036052$ | $16902759321$ | $6117677843949$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $8$ | $352$ | $6860$ | $129700$ | $2470688$ | $47026222$ | $893856608$ | $16983689284$ | $322687697780$ | $6131061446752$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=10x^6+7x^5+17x^4+5x^3+14x^2+10x+15$
- $y^2=2x^6+2x^3+4$
- $y^2=x^6+x^3+14$
- $y^2=13x^6+13x^5+14x^4+6x^3+9x^2+3x+13$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19^{6}}$.
Endomorphism algebra over $\F_{19}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{7})\). |
The base change of $A$ to $\F_{19^{6}}$ is 1.47045881.aooc 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-21}) \)$)$ |
- Endomorphism algebra over $\F_{19^{2}}$
The base change of $A$ to $\F_{19^{2}}$ is the simple isogeny class 2.361.ak_akb and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{7})\). - Endomorphism algebra over $\F_{19^{3}}$
The base change of $A$ to $\F_{19^{3}}$ is the simple isogeny class 2.6859.a_aooc and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{7})\).
Base change
This is a primitive isogeny class.