Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 9 x + 45 x^{2} - 171 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.116535216599$, $\pm0.468549984315$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.2058997.1 |
Galois group: | $D_{4}$ |
Jacobians: | $7$ |
Isomorphism classes: | 7 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $227$ | $133249$ | $46870733$ | $16887045517$ | $6129376351472$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $11$ | $371$ | $6833$ | $129579$ | $2475416$ | $47065295$ | $893959931$ | $16983634435$ | $322687938269$ | $6131073282086$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=18x^6+13x^5+6x^4+5x^3+x^2+8x+13$
- $y^2=13x^6+14x^5+17x^4+5x^3+11x^2+5x+17$
- $y^2=14x^6+15x^5+4x^4+18x^3+12x+10$
- $y^2=18x^6+8x^5+2x^4+5x^3+13x^2+8x+13$
- $y^2=14x^6+16x^5+16x^4+x^3+17x^2+6x+14$
- $y^2=12x^6+x^5+6x^4+14x^3+5x^2+12x+3$
- $y^2=10x^6+11x^5+12x^3+5x^2+4x+1$
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 4.0.2058997.1. |
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.j_bt | $2$ | (not in LMFDB) |