Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 8 x + 19 x^{2} )( 1 - 5 x + 19 x^{2} )$ |
$1 - 13 x + 78 x^{2} - 247 x^{3} + 361 x^{4}$ | |
Frobenius angles: | $\pm0.130073469147$, $\pm0.305569972467$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $4$ |
Isomorphism classes: | 16 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $180$ | $126000$ | $47764080$ | $17061912000$ | $6133935663900$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $7$ | $349$ | $6964$ | $130921$ | $2477257$ | $47044582$ | $893875003$ | $16983776401$ | $322689478156$ | $6131074035949$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=12x^6+8x^5+5x^4+18x^2+4x+4$
- $y^2=2x^6+10x^5+11x^4+2x^3+7x^2+11x+10$
- $y^2=16x^5+18x^4+3x^3+3x^2+12x+8$
- $y^2=14x^6+2x^5+3x^4+16x^3+x^2+18x+17$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19}$.
Endomorphism algebra over $\F_{19}$The isogeny class factors as 1.19.ai $\times$ 1.19.af and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.