Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 11 x + 58 x^{2} - 209 x^{3} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.0194296767276$, $\pm0.415073966325$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-13 +2 \sqrt{41}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 2 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $200$ | $128000$ | $46738400$ | $16847360000$ | $6116072751000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $9$ | $357$ | $6816$ | $129273$ | $2470039$ | $47032662$ | $893874501$ | $16983472113$ | $322685962464$ | $6131057792677$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):
- $y^2=14 x^5+6 x^4+8 x^2+9 x+14$
- $y^2=2 x^6+3 x^5+11 x^4+9 x^3+16 x^2+18 x+10$
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 \(\Q(\sqrt{-13 +2 \sqrt{41}})\). |
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.l_cg | $2$ | (not in LMFDB) |