Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 20 x^{2} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.161786489280$, $\pm0.838213510720$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-2}, \sqrt{-29})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $18$ |
| Isomorphism classes: | 54 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $342$ | $116964$ | $47059542$ | $17067854736$ | $6131064465702$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $322$ | $6860$ | $130966$ | $2476100$ | $47073202$ | $893871740$ | $16983876958$ | $322687697780$ | $6131062673602$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):
- $y^2=12 x^6+7 x^5+11 x^4+x^3+6 x^2+17 x+16$
- $y^2=5 x^6+14 x^5+3 x^4+2 x^3+12 x^2+15 x+13$
- $y^2=6 x^6+12 x^5+7 x^4+2 x^3+8 x^2+17 x+9$
- $y^2=12 x^6+5 x^5+14 x^4+4 x^3+16 x^2+15 x+18$
- $y^2=17 x^6+8 x^5+4 x^4+6 x^3+4 x^2+6$
- $y^2=15 x^6+16 x^5+8 x^4+12 x^3+8 x^2+12$
- $y^2=15 x^6+2 x^5+16 x^4+12 x^3+4 x^2+13$
- $y^2=11 x^6+4 x^5+13 x^4+5 x^3+8 x^2+7$
- $y^2=12 x^6+18 x^5+16 x^4+12 x^3+7 x^2+9 x+12$
- $y^2=5 x^6+17 x^5+13 x^4+5 x^3+14 x^2+18 x+5$
- $y^2=14 x^6+16 x^5+12 x^4+15 x^3+12 x^2+11 x+10$
- $y^2=9 x^6+13 x^5+5 x^4+11 x^3+5 x^2+3 x+1$
- $y^2=6 x^6+6 x^5+4 x^4+7 x^3+11 x^2+11 x+13$
- $y^2=12 x^6+12 x^5+8 x^4+14 x^3+3 x^2+3 x+7$
- $y^2=4 x^6+2 x^5+11 x^4+14 x^3+16 x^2+8 x+18$
- $y^2=8 x^6+4 x^5+3 x^4+9 x^3+13 x^2+16 x+17$
- $y^2=12 x^6+6 x^5+x^4+2 x^3+17 x^2+15 x+9$
- $y^2=5 x^6+12 x^5+2 x^4+4 x^3+15 x^2+11 x+18$
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 \(\Q(\sqrt{-2}, \sqrt{-29})\). |
| The base change of $A$ to $\F_{19^{2}}$ is 1.361.au 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-29}) \)$)$ |
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_u | $4$ | (not in LMFDB) |
| 2.19.ag_s | $8$ | (not in LMFDB) |
| 2.19.g_s | $8$ | (not in LMFDB) |