Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 2 x - 8 x^{2} + 46 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.266750468722$, $\pm0.840756633373$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-36 +2 \sqrt{55}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $28$ |
| Isomorphism classes: | 56 |
| Cyclic group of points: | yes |
This isogeny class is simple and 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $570$ | $270180$ | $150423570$ | $78725048400$ | $41416614689250$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $26$ | $510$ | $12362$ | $281318$ | $6434806$ | $148052430$ | $3404609542$ | $78310839358$ | $1801151278826$ | $41426513993550$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):
- $y^2=6 x^6+22 x^5+2 x^4+13 x^3+9 x^2+10 x+19$
- $y^2=9 x^6+x^5+9 x^4+11 x^3+20 x^2+2 x+20$
- $y^2=3 x^6+22 x^5+17 x^4+3 x^3+13 x^2+9 x$
- $y^2=20 x^5+11 x^4+3 x^3+8 x^2+15 x+2$
- $y^2=4 x^6+12 x^5+21 x^4+2 x^3+5 x^2+22 x+15$
- $y^2=10 x^6+8 x^5+18 x^4+15 x^3+20 x^2+14 x+12$
- $y^2=3 x^6+3 x^5+8 x^4+17 x^3+17 x^2+5 x+17$
- $y^2=5 x^6+7 x^5+10 x^4+2 x^3+12 x^2+22 x+2$
- $y^2=4 x^6+17 x^5+6 x^4+20 x^3+19 x^2+12 x+8$
- $y^2=6 x^6+4 x^5+17 x^4+18 x^3+16 x^2+19 x+13$
- $y^2=18 x^6+20 x^5+18 x^4+5 x^3+6 x^2+13 x+1$
- $y^2=11 x^6+22 x^5+4 x^4+12 x^3+22 x^2+2 x+20$
- $y^2=16 x^6+9 x^5+15 x^4+4 x^3+21 x^2+21 x+21$
- $y^2=16 x^6+15 x^5+14 x^4+14 x^3+16 x^2+8 x+18$
- $y^2=19 x^6+20 x^5+20 x^4+17 x^3+13 x^2+6 x+21$
- $y^2=3 x^6+6 x^5+20 x^4+4 x^3+18 x^2+7 x+2$
- $y^2=11 x^6+21 x^5+13 x^4+11 x^3+12 x^2+4 x+18$
- $y^2=9 x^5+14 x^3+6 x^2+5 x+8$
- $y^2=11 x^6+10 x^5+11 x^4+7 x^2+18 x+5$
- $y^2=9 x^6+11 x^5+17 x^4+19 x^2+22 x+15$
- $y^2=14 x^6+5 x^5+11 x^4+10 x^3+21 x^2+4 x+22$
- $y^2=8 x^6+13 x^5+12 x^4+3 x^3+20 x^2+7 x+20$
- $y^2=2 x^6+3 x^5+15 x^4+2 x^3+19 x^2+7 x+3$
- $y^2=15 x^6+18 x^5+17 x^4+9 x^3+21 x^2+22 x+11$
- $y^2=2 x^6+2 x^5+11 x^4+10 x^3+6 x^2+8 x+19$
- $y^2=12 x^6+7 x^5+15 x^4+x^3+5 x^2+20 x+17$
- $y^2=19 x^6+9 x^5+15 x^4+3 x^3+9 x^2+8 x+15$
- $y^2=10 x^6+17 x^5+4 x^4+13 x^3+15 x^2+16 x+7$
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 \(\Q(\sqrt{-36 +2 \sqrt{55}})\). |
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.ac_ai | $2$ | (not in LMFDB) |