Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 2 x + 33 x^{2} - 38 x^{3} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.370507689379$, $\pm0.553171643954$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-69 -2 \sqrt{6}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $12$ |
| Isomorphism classes: | 12 |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $355$ | $154425$ | $47568580$ | $16915251225$ | $6129871160275$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $424$ | $6936$ | $129796$ | $2475618$ | $47042782$ | $893834022$ | $16983761476$ | $322689395544$ | $6131062790824$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=6 x^6+6 x^4+3 x^3+12 x^2+17 x+4$
- $y^2=x^6+9 x^5+15 x^4+14 x^3+11 x^2+18 x+11$
- $y^2=17 x^6+3 x^5+15 x^4+5 x^3+8 x+4$
- $y^2=12 x^6+11 x^5+11 x^4+14 x^3+7 x^2+3 x+5$
- $y^2=x^6+7 x^5+2 x^4+15 x^3+4 x^2+6 x+12$
- $y^2=15 x^6+4 x^5+7 x^4+4 x^3+13 x^2+6 x+3$
- $y^2=17 x^6+10 x^5+6 x^3+11 x^2+14 x+14$
- $y^2=x^6+18 x^5+17 x^4+16 x^3+12 x^2+3 x+16$
- $y^2=x^6+13 x^5+18 x^4+7 x^2+4 x+8$
- $y^2=13 x^6+11 x^5+17 x^4+18 x^3+2 x^2+5 x+3$
- $y^2=12 x^6+15 x^4+13 x^3+18 x^2+16 x+10$
- $y^2=12 x^6+12 x^5+4 x^4+7 x^3+4 x^2+16 x+14$
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 \(\Q(\sqrt{-69 -2 \sqrt{6}})\). |
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.c_bh | $2$ | (not in LMFDB) |