Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 6 x^{2} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.229181340406$, $\pm0.770818659594$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-10}, \sqrt{13})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $22$ |
| Isomorphism classes: | 40 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $524$ | $274576$ | $148045196$ | $78884586496$ | $41426503381964$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $24$ | $518$ | $12168$ | $281886$ | $6436344$ | $148054502$ | $3404825448$ | $78310015678$ | $1801152661464$ | $41426495550278$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 22 curves (of which all are hyperelliptic):
- $y^2=14 x^6+5 x^5+15 x^4+19 x^3+6 x^2+10 x+2$
- $y^2=13 x^6+5 x^5+2 x^4+16 x^3+3 x+16$
- $y^2=19 x^6+2 x^5+10 x^4+11 x^3+15 x+11$
- $y^2=3 x^6+15 x^5+20 x^4+4 x^3+16 x^2+3 x+5$
- $y^2=15 x^6+6 x^5+8 x^4+20 x^3+11 x^2+15 x+2$
- $y^2=4 x^6+22 x^5+12 x^4+18 x^3+22 x^2+18 x+2$
- $y^2=20 x^6+18 x^5+14 x^4+21 x^3+18 x^2+21 x+10$
- $y^2=8 x^6+x^5+8 x^4+11 x^3+15 x^2+x+15$
- $y^2=19 x^6+8 x^5+3 x^4+5 x^2+12 x+13$
- $y^2=16 x^6+21 x^5+15 x^4+20 x^3+18 x^2+10 x+19$
- $y^2=15 x^6+13 x^5+15 x^4+17 x^3+8 x^2+13 x+8$
- $y^2=4 x^6+7 x^5+22 x^4+8 x^3+11 x^2+7 x+14$
- $y^2=20 x^6+12 x^5+18 x^4+17 x^3+9 x^2+12 x+1$
- $y^2=21 x^6+20 x^5+7 x^4+8 x^3+13 x^2+15 x+12$
- $y^2=7 x^6+20 x^5+21 x^4+15 x^3+18 x^2+10 x+3$
- $y^2=11 x^6+20 x^5+6 x^4+4 x^3+17 x^2+20 x+12$
- $y^2=5 x^6+12 x^5+18 x^4+2 x^3+20 x^2+14 x+6$
- $y^2=2 x^6+14 x^5+21 x^4+10 x^3+8 x^2+x+7$
- $y^2=16 x^6+5 x^5+18 x^4+10 x^3+6 x^2+5 x+18$
- $y^2=11 x^6+2 x^5+21 x^4+4 x^3+7 x^2+2 x+21$
- $y^2=14 x^6+2 x^5+6 x^4+12 x^3+14 x^2+7 x+14$
- $y^2=x^6+10 x^5+7 x^4+14 x^3+x^2+12 x+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23^{2}}$.
Endomorphism algebra over $\F_{23}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-10}, \sqrt{13})\). |
| The base change of $A$ to $\F_{23^{2}}$ is 1.529.ag 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-130}) \)$)$ |
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 |
|---|---|---|
| 2.23.a_g | $4$ | (not in LMFDB) |