Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 7 x^{2} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.274313691880$, $\pm0.725686308120$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{39}, \sqrt{-53})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $20$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $537$ | $288369$ | $148025124$ | $78877284201$ | $41426520117657$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $24$ | $544$ | $12168$ | $281860$ | $6436344$ | $148014358$ | $3404825448$ | $78310068484$ | $1801152661464$ | $41426529021664$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):
- $y^2=4 x^6+10 x^5+x^4+20 x^3+6 x^2+16 x+6$
- $y^2=20 x^6+4 x^5+5 x^4+8 x^3+7 x^2+11 x+7$
- $y^2=6 x^6+15 x^5+2 x^4+5 x^3+6 x^2+2 x+17$
- $y^2=7 x^6+6 x^5+10 x^4+2 x^3+7 x^2+10 x+16$
- $y^2=x^6+16 x^5+19 x^4+21 x^3+2 x^2+7 x+15$
- $y^2=5 x^6+11 x^5+3 x^4+13 x^3+10 x^2+12 x+6$
- $y^2=16 x^6+2 x^5+21 x^4+19 x^3+19 x^2+10 x+12$
- $y^2=11 x^6+10 x^5+13 x^4+3 x^3+3 x^2+4 x+14$
- $y^2=20 x^6+9 x^5+8 x^4+16 x^3+11 x^2+15 x+21$
- $y^2=8 x^6+18 x^5+18 x^3+20 x^2+14 x+20$
- $y^2=17 x^6+21 x^5+21 x^3+8 x^2+x+8$
- $y^2=12 x^6+10 x^5+16 x^4+17 x^3+22 x^2+13 x+6$
- $y^2=14 x^6+4 x^5+11 x^4+16 x^3+18 x^2+19 x+7$
- $y^2=20 x^6+13 x^5+22 x^4+13 x^3+18 x^2+7 x+20$
- $y^2=10 x^6+20 x^5+13 x^4+8 x^3+15 x^2+9$
- $y^2=21 x^6+4 x^5+2 x^4+12 x^3+19 x^2+20 x+12$
- $y^2=10 x^6+x^5+10 x^4+21 x^3+3 x^2+22 x+5$
- $y^2=4 x^6+5 x^5+4 x^4+13 x^3+15 x^2+18 x+2$
- $y^2=8 x^6+12 x^5+22 x^4+8 x^3+x^2+3 x+2$
- $y^2=17 x^6+14 x^5+18 x^4+17 x^3+5 x^2+15 x+10$
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 \(\Q(\sqrt{39}, \sqrt{-53})\). |
| The base change of $A$ to $\F_{23^{2}}$ is 1.529.h 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2067}) \)$)$ |
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_ah | $4$ | (not in LMFDB) |