Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 12 x + 80 x^{2} - 276 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.218762671529$, $\pm0.341325181820$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-54 +12 \sqrt{2}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $322$ | $289156$ | $152025538$ | $78707106576$ | $41438017377442$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $546$ | $12492$ | $281254$ | $6438132$ | $148025346$ | $3404774196$ | $78310960318$ | $1801152606060$ | $41426503978146$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=11 x^6+13 x^5+5 x^4+12 x^3+17 x^2+19 x$
- $y^2=17 x^6+16 x^5+19 x^4+17 x^3+5 x^2+18 x+14$
- $y^2=7 x^6+16 x^5+2 x^4+18 x^3+8 x^2+22 x+21$
- $y^2=12 x^6+22 x^5+20 x^4+9 x^3+15 x^2+7$
- $y^2=17 x^6+15 x^5+15 x^4+19 x^3+17 x^2+6 x+5$
- $y^2=2 x^6+6 x^5+14 x^4+6 x^3+11 x^2+2 x+3$
- $y^2=5 x^6+18 x^5+17 x^4+5 x^3+6 x^2+19 x+6$
- $y^2=7 x^6+6 x^5+5 x^4+16 x^3+x^2+19 x+17$
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 \(\Q(\sqrt{-54 +12 \sqrt{2}})\). |
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.m_dc | $2$ | (not in LMFDB) |