Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 12 x + 77 x^{2} - 276 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.171291584331$, $\pm0.371639008133$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-186 -30 \sqrt{5}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $16$ |
| Isomorphism classes: | 16 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $319$ | $285505$ | $150690496$ | $78486752025$ | $41426425940119$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $540$ | $12384$ | $280468$ | $6436332$ | $148041870$ | $3404956644$ | $78311764708$ | $1801153569312$ | $41426495138700$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):
- $y^2=14 x^6+18 x^5+20 x^4+10 x^3+2 x^2+5 x+7$
- $y^2=19 x^6+6 x^4+10 x^3+12 x^2+6 x+7$
- $y^2=6 x^6+13 x^5+20 x^4+10 x^3+19 x^2+22 x+19$
- $y^2=13 x^6+x^5+8 x^4+19 x^3+14 x^2+12 x+11$
- $y^2=19 x^6+2 x^5+14 x^4+2 x^3+19 x^2+21 x+20$
- $y^2=10 x^6+20 x^5+x^4+20 x^3+5 x^2+21 x+20$
- $y^2=7 x^6+10 x^5+2 x^4+19 x^2+4 x+7$
- $y^2=19 x^6+17 x^5+6 x^4+19 x^3+5 x^2+17 x+10$
- $y^2=19 x^6+22 x^5+8 x^4+19 x^3+6 x+12$
- $y^2=14 x^6+4 x^5+4 x^4+8 x^3+7 x^2+6 x+14$
- $y^2=14 x^6+3 x^5+4 x^4+21 x^3+16 x^2+19 x+7$
- $y^2=7 x^6+17 x^5+17 x^4+6 x^3+2 x^2+9 x+22$
- $y^2=4 x^6+18 x^5+8 x^4+6 x^3+5 x^2+11 x+22$
- $y^2=19 x^6+21 x^5+x^4+14 x^2+x+5$
- $y^2=15 x^6+7 x^5+3 x^4+11 x^3+17 x^2+x+18$
- $y^2=10 x^6+3 x^5+10 x^4+11 x^3+18 x^2+18 x+11$
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 \(\Q(\sqrt{-186 -30 \sqrt{5}})\). |
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.m_cz | $2$ | (not in LMFDB) |