Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 13 x + 79 x^{2} - 247 x^{3} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.161616782251$, $\pm0.288204697885$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-130 +26 \sqrt{5}})\) |
| Galois group: | $C_4$ |
| Jacobians: | $2$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $181$ | $126881$ | $48039391$ | $17109268445$ | $6138926902096$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $351$ | $7003$ | $131283$ | $2479272$ | $47050491$ | $893866813$ | $16983571443$ | $322688189557$ | $6131069106406$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):
- $y^2=18 x^6+2 x^5+16 x^4+2 x^3+11 x^2+13 x+3$
- $y^2=15 x^6+5 x^5+18 x^4+10 x^3+11 x^2+15$
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 \(\Q(\sqrt{-130 +26 \sqrt{5}})\). |
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.n_db | $2$ | (not in LMFDB) |