Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 8 x + 19 x^{2} )( 1 - x + 19 x^{2} )$ |
$1 - 9 x + 46 x^{2} - 171 x^{3} + 361 x^{4}$ | |
Frobenius angles: | $\pm0.130073469147$, $\pm0.463406802480$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $10$ |
Isomorphism classes: | 86 |
This isogeny class is not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $228$ | $134064$ | $47056464$ | $16905470400$ | $6130377927948$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $11$ | $373$ | $6860$ | $129721$ | $2475821$ | $47067046$ | $893972279$ | $16983663601$ | $322687697780$ | $6131071184053$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=10x^6+18x^5+11x^4+5x^3+9x^2+15x+3$
- $y^2=6x^6+15x^5+9x^4+4x^3+18x^2+14x+12$
- $y^2=3x^6+16x^5+11x^4+18x^3+7x$
- $y^2=x^6+2x^3+15$
- $y^2=6x^6+13x^5+2x^4+16x^3+4x^2+6x+18$
- $y^2=12x^6+2x^5+5x^4+9x^3+16x^2+15x+17$
- $y^2=x^6+9x^5+2x^4+3x^3+11x^2+7x$
- $y^2=17x^6+11x^5+18x^4+9x^2+13x+2$
- $y^2=12x^6+10x^5+5x^4+10x^3+13x^2+8x+3$
- $y^2=15x^6+12x^5+12x^4+9x^2+4x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19^{6}}$.
Endomorphism algebra over $\F_{19}$The isogeny class factors as 1.19.ai $\times$ 1.19.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
The base change of $A$ to $\F_{19^{6}}$ is 1.47045881.pra 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
- Endomorphism algebra over $\F_{19^{2}}$
The base change of $A$ to $\F_{19^{2}}$ is 1.361.aba $\times$ 1.361.bl. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{19^{3}}$
The base change of $A$ to $\F_{19^{3}}$ is 1.6859.ace $\times$ 1.6859.ce. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.