Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 10 x + 57 x^{2} - 190 x^{3} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.173854422602$, $\pm0.405491683409$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-45 -10 \sqrt{6}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $1$ |
| Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $219$ | $135561$ | $48021444$ | $17002738425$ | $6130443137979$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $376$ | $7000$ | $130468$ | $2475850$ | $47056606$ | $893970430$ | $16983838468$ | $322686954520$ | $6131058005656$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=12 x^6+3 x^5+13 x^4+6 x^3+5 x^2+8 x+15$
- $y^2=12 x^6+8 x^5+2 x^4+x^3+2 x^2+6 x+14$
- $y^2=18 x^6+4 x^5+12 x^4+6 x^3+6 x^2+6 x+8$
- $y^2=17 x^6+16 x^5+16 x^4+9 x^3+3 x^2+12 x+11$
- $y^2=3 x^6+x^5+13 x^4+13 x^3+12 x^2+11 x+18$
- $y^2=14 x^6+9 x^5+4 x^3+2 x^2+11 x+10$
- $y^2=15 x^6+2 x^5+3 x^4+17 x^3+5 x^2+10$
- $y^2=12 x^6+17 x^5+12 x^4+17 x^3+3 x^2+15 x+17$
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 \(\Q(\sqrt{-45 -10 \sqrt{6}})\). |
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.k_cf | $2$ | (not in LMFDB) |