Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 8 x + 54 x^{2} - 184 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.247829138847$, $\pm0.461022733442$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-34 +8 \sqrt{2}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $28$ |
| Isomorphism classes: | 44 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $392$ | $304192$ | $150895304$ | $78345257984$ | $41431610857672$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $16$ | $574$ | $12400$ | $279966$ | $6437136$ | $148054942$ | $3404835184$ | $78310114494$ | $1801148256016$ | $41426514705534$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):
- $y^2=2 x^6+16 x^5+10 x^4+12 x+16$
- $y^2=11 x^6+3 x^5+6 x^4+15 x^3+10 x^2+7 x+7$
- $y^2=17 x^6+22 x^5+16 x^3+13 x^2+11 x+17$
- $y^2=22 x^6+6 x^5+9 x^4+12 x^3+22 x^2+9 x+19$
- $y^2=14 x^6+15 x^5+x^4+3 x^3+22 x^2+14 x+15$
- $y^2=7 x^6+19 x^5+21 x^4+18 x^3+2 x^2+16 x+18$
- $y^2=15 x^6+20 x^5+3 x^4+15 x^3+3 x^2+18 x+11$
- $y^2=10 x^6+17 x^5+3 x^4+17 x^3+20 x+5$
- $y^2=18 x^6+5 x^5+12 x^4+14 x^3+21 x^2+9 x+1$
- $y^2=11 x^6+12 x^5+x^4+20 x^3+x^2+22 x+22$
- $y^2=11 x^6+20 x^5+2 x^4+12 x^3+3 x^2+5 x+19$
- $y^2=16 x^6+4 x^5+4 x^4+20 x^3+21 x^2+11 x+8$
- $y^2=15 x^6+8 x^5+17 x^4+18 x^3+14 x^2+20 x+18$
- $y^2=19 x^6+16 x^5+19 x^4+7 x^3+x^2+9 x+21$
- $y^2=20 x^5+19 x^4+19 x^3+3 x^2+20 x+21$
- $y^2=6 x^6+8 x^5+3 x^4+15 x^3+2 x^2+15 x+3$
- $y^2=5 x^6+12 x^5+8 x^3+7 x^2+17 x+19$
- $y^2=11 x^6+20 x^5+10 x^4+5 x^3+3 x^2+2 x+14$
- $y^2=7 x^6+18 x^5+21 x^4+4 x^3+15 x^2+10 x+6$
- $y^2=19 x^6+13 x^5+13 x^4+14 x^3+2 x^2+8 x+22$
- $y^2=3 x^6+4 x^5+9 x^4+3 x^3+17 x^2+12 x+12$
- $y^2=22 x^6+19 x^5+18 x^4+14 x^3+8 x^2+13 x+21$
- $y^2=20 x^6+18 x^5+4 x^4+18 x^3+10 x^2+7 x+9$
- $y^2=15 x^6+7 x^5+7 x^4+6 x^3+16 x^2+13 x+15$
- $y^2=8 x^6+7 x^5+16 x^4+11 x^3+3 x^2+9 x+14$
- $y^2=11 x^6+9 x^5+16 x^4+13 x^3+6 x^2+16 x+20$
- $y^2=12 x^6+7 x^5+18 x^4+19 x^3+11 x^2+14 x+22$
- $y^2=14 x^6+2 x^5+14 x^4+5 x^3+16 x^2+16 x+7$
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 \(\Q(\sqrt{-34 +8 \sqrt{2}})\). |
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.i_cc | $2$ | (not in LMFDB) |