Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 7 x^{2} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.279486410696$, $\pm0.720513589304$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-5}, \sqrt{31})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $30$ |
| Isomorphism classes: | 36 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
This isogeny class is simple but not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $369$ | $136161$ | $47038644$ | $17159690025$ | $6131070216729$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $376$ | $6860$ | $131668$ | $2476100$ | $47031406$ | $893871740$ | $16983178468$ | $322687697780$ | $6131074175656$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):
- $y^2=18 x^6+5 x^5+10 x^4+11 x^3+12 x^2+2 x+6$
- $y^2=8 x^6+2 x^5+2 x^4+13 x^3+14 x^2+4 x+12$
- $y^2=16 x^6+4 x^5+4 x^4+7 x^3+9 x^2+8 x+5$
- $y^2=13 x^6+18 x^5+5 x^4+5 x^3+11 x^2+11 x+8$
- $y^2=14 x^6+x^5+x^4+11 x^3+4 x^2+7 x+17$
- $y^2=9 x^6+2 x^5+2 x^4+3 x^3+8 x^2+14 x+15$
- $y^2=3 x^6+16 x^5+3 x^4+18 x^2+10 x+18$
- $y^2=6 x^6+13 x^5+6 x^4+17 x^2+x+17$
- $y^2=3 x^6+17 x^5+15 x^4+15 x^3+8 x^2+7 x+18$
- $y^2=3 x^6+9 x^5+2 x^4+11 x^3+10 x+14$
- $y^2=6 x^6+18 x^5+4 x^4+3 x^3+x+9$
- $y^2=7 x^6+5 x^5+13 x^4+5 x^3+17 x^2+3 x+12$
- $y^2=7 x^6+4 x^5+6 x^4+11 x^3+12 x^2+16 x+4$
- $y^2=14 x^6+8 x^5+12 x^4+3 x^3+5 x^2+13 x+8$
- $y^2=5 x^6+3 x^5+x^4+3 x^3+10 x^2+16 x+13$
- $y^2=13 x^6+11 x^5+13 x^4+5 x^3+x^2+x+4$
- $y^2=7 x^6+3 x^5+7 x^4+10 x^3+2 x^2+2 x+8$
- $y^2=15 x^6+12 x^5+5 x^4+6 x^3+2 x^2+5 x+3$
- $y^2=2 x^6+16 x^5+5 x^4+3 x^3+18 x^2+14 x+3$
- $y^2=4 x^6+13 x^5+10 x^4+6 x^3+17 x^2+9 x+6$
- $y^2=13 x^6+3 x^5+11 x^4+8 x^3+11 x^2+11 x+3$
- $y^2=7 x^6+6 x^5+3 x^4+16 x^3+3 x^2+3 x+6$
- $y^2=13 x^6+14 x^5+7 x^4+2 x^3+8 x^2+x+4$
- $y^2=7 x^6+9 x^5+14 x^4+4 x^3+16 x^2+2 x+8$
- $y^2=14 x^6+18 x^5+2 x^4+15 x^3+10 x^2+14 x+5$
- $y^2=9 x^6+17 x^5+4 x^4+11 x^3+x^2+9 x+10$
- $y^2=2 x^6+8 x^5+4 x^4+6 x^3+13 x^2+7 x+4$
- $y^2=4 x^6+16 x^5+8 x^4+12 x^3+7 x^2+14 x+8$
- $y^2=16 x^6+3 x^5+4 x^4+2 x^3+8 x^2+8 x+7$
- $y^2=13 x^6+6 x^5+8 x^4+4 x^3+16 x^2+16 x+14$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19^{2}}$.
Endomorphism algebra over $\F_{19}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-5}, \sqrt{31})\). |
| The base change of $A$ to $\F_{19^{2}}$ is 1.361.h 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-155}) \)$)$ |
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.a_ah | $4$ | (not in LMFDB) |