Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 10 x + 58 x^{2} - 230 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.145604305875$, $\pm0.453559101412$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-54 +10 \sqrt{13}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $30$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $348$ | $288144$ | $148664556$ | $78140042496$ | $41426191001868$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $14$ | $546$ | $12218$ | $279230$ | $6436294$ | $148073922$ | $3405041458$ | $78311251774$ | $1801151571614$ | $41426514421986$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):
- $y^2=7 x^6+7 x^5+x^4+3 x^3+3 x^2+3 x+18$
- $y^2=7 x^6+20 x^5+8 x^4+19 x^3+22 x^2+14 x+9$
- $y^2=20 x^6+13 x^5+10 x^4+7 x^3+19 x^2+15 x+20$
- $y^2=5 x^6+21 x^5+2 x^4+8 x^3+19 x^2+16 x$
- $y^2=22 x^6+5 x^5+16 x^4+4 x^3+6 x^2+14 x+19$
- $y^2=5 x^6+22 x^5+20 x^4+15 x^3+12 x^2+6 x+17$
- $y^2=17 x^6+13 x^4+3 x^3+12 x^2+6 x+7$
- $y^2=15 x^5+2 x^4+4 x^3+8 x^2+6 x+14$
- $y^2=22 x^6+5 x^5+16 x^4+15 x^3+11 x^2+6 x+21$
- $y^2=14 x^6+19 x^5+11 x^4+9 x^3+11 x^2+19 x+7$
- $y^2=18 x^6+18 x^5+13 x^4+17 x^3+8 x^2+20 x+15$
- $y^2=15 x^6+20 x^5+9 x^4+21 x^3+5 x+6$
- $y^2=21 x^6+10 x^5+12 x^4+9 x^3+2 x^2+4 x+5$
- $y^2=17 x^6+9 x^5+x^4+14 x^3+11 x^2+20 x+16$
- $y^2=10 x^6+14 x^5+22 x^4+8 x^3+17 x^2+14 x+15$
- $y^2=18 x^6+10 x^5+3 x^4+x^3+5 x^2+3 x+21$
- $y^2=10 x^6+6 x^5+2 x^4+7 x^3+16 x^2+13 x+14$
- $y^2=14 x^6+4 x^5+7 x^4+8 x^3+14 x^2+7 x+15$
- $y^2=11 x^6+12 x^5+12 x^4+12 x^3+11 x^2+9 x+9$
- $y^2=21 x^5+x^4+13 x^3+21 x^2+13 x+22$
- $y^2=21 x^6+8 x^5+10 x^4+x^3+20 x^2+5 x+14$
- $y^2=19 x^6+6 x^5+7 x^4+19 x^3+12 x^2+4 x+16$
- $y^2=10 x^6+14 x^4+21 x^3+x^2+16 x+19$
- $y^2=21 x^6+18 x^5+10 x^4+12 x^3+16 x^2+17 x+17$
- $y^2=17 x^6+10 x^5+x^4+17 x^3+12 x^2+13 x+20$
- $y^2=13 x^6+8 x^5+15 x^4+14 x^3+19 x^2+14 x+7$
- $y^2=9 x^6+17 x^5+4 x^4+22 x^3+21 x^2+15 x+15$
- $y^2=21 x^6+20 x^5+4 x^4+8 x^3+21 x^2+15 x+20$
- $y^2=3 x^6+17 x^5+9 x^4+2 x^3+13 x^2+5 x+19$
- $y^2=4 x^6+22 x^5+14 x^4+12 x^2+18 x+7$
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 \(\Q(\sqrt{-54 +10 \sqrt{13}})\). |
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.k_cg | $2$ | (not in LMFDB) |