Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 10 x + 53 x^{2} - 190 x^{3} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.114247126664$, $\pm0.432392726124$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-41 -10 \sqrt{10}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $215$ | $132225$ | $47189060$ | $16909065225$ | $6125993285375$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $368$ | $6880$ | $129748$ | $2474050$ | $47057438$ | $893979670$ | $16983847588$ | $322687758640$ | $6131068120448$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=8 x^6+13 x^5+18 x^4+6 x^3+7 x+18$
- $y^2=3 x^6+x^5+17 x^4+13 x^3+x^2+2 x+3$
- $y^2=18 x^6+18 x^5+14 x^4+13 x^3+12 x^2+17 x+2$
- $y^2=2 x^6+5 x^5+13 x^4+6 x^3+x^2+11 x+18$
- $y^2=14 x^6+3 x^5+14 x^4+11 x^3+18 x^2+3 x+17$
- $y^2=15 x^6+9 x^5+15 x^4+12 x^3+8 x^2+5 x+10$
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 \(\Q(\sqrt{-41 -10 \sqrt{10}})\). |
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_cb | $2$ | (not in LMFDB) |