Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 11 x + 69 x^{2} - 253 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.174086733330$, $\pm0.405448433343$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-218 -22 \sqrt{29}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $1$ |
| Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $335$ | $289105$ | $150338285$ | $78372028925$ | $41423951738800$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $13$ | $547$ | $12355$ | $280059$ | $6435948$ | $148054879$ | $3405017629$ | $78311574051$ | $1801150891135$ | $41426489691582$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=20 x^6+14 x^5+10 x^4+4 x^3+8 x^2+2 x+19$
- $y^2=16 x^6+2 x^5+4 x^4+16 x^3+4 x^2+9 x+17$
- $y^2=21 x^5+4 x^4+15 x^3+14 x^2+15 x+21$
- $y^2=4 x^6+14 x^5+14 x^4+6 x^3+5 x^2+18 x+22$
- $y^2=8 x^6+5 x^4+17 x^3+4 x^2+14 x+8$
- $y^2=9 x^6+12 x^5+4 x^4+20 x^3+18 x^2+14 x+3$
- $y^2=7 x^6+19 x^5+4 x^4+16 x^2+15 x+4$
- $y^2=14 x^6+12 x^5+7 x^4+22 x^3+3 x^2+6 x+6$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23}$.
Endomorphism algebra over $\F_{23}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-218 -22 \sqrt{29}})\). |
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 |
|---|---|---|
| 2.23.l_cr | $2$ | (not in LMFDB) |