Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 12 x + 69 x^{2} - 228 x^{3} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.106312411237$, $\pm0.357895350441$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.12625.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 8 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $191$ | $128161$ | $47539136$ | $16981973305$ | $6125692347071$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $356$ | $6932$ | $130308$ | $2473928$ | $47039366$ | $893914232$ | $16984032388$ | $322689696428$ | $6131069897156$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=10x^6+17x^5+3x^4+6x^3+8x^2+7x+2$
- $y^2=13x^6+11x^4+14x^3+17x^2+18x+18$
- $y^2=14x^6+8x^5+2x^4+8x^3+6x^2+5x+13$
- $y^2=14x^6+9x^5+15x^4+5x^3+12x^2+x+14$
- $y^2=13x^6+8x^5+x^4+15x^3+11x^2+9x+8$
- $y^2=11x^6+2x^5+2x^4+18x^3+12x^2+18x+8$
- $y^2=10x^6+17x^5+7x^4+7x^3+13x+13$
- $y^2=3x^6+5x^5+9x^4+11x^3+10x^2+3x+5$
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 4.0.12625.1. |
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.m_cr | $2$ | (not in LMFDB) |