Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 13 x + 77 x^{2} - 247 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.0986133210333$, $\pm0.318874605641$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.64389.1 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
Isomorphism classes: | 2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $179$ | $125121$ | $47489237$ | $17014078701$ | $6128624886224$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $7$ | $347$ | $6925$ | $130555$ | $2475112$ | $47036567$ | $893860807$ | $16983810019$ | $322689806185$ | $6131075312102$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=8x^6+10x^5+7x^4+16x^3+9x^2+12x+10$
- $y^2=15x^6+6x^5+11x^4+x^3+18x^2+x+15$
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 4.0.64389.1. |
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.n_cz | $2$ | (not in LMFDB) |