Invariants
| Base field: | $\F_{43}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 20 x + 183 x^{2} - 860 x^{3} + 1849 x^{4}$ |
| Frobenius angles: | $\pm0.147488713818$, $\pm0.282880549156$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-69 +20 \sqrt{3}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1153$ | $3358689$ | $6353274436$ | $11703314874921$ | $21614849206697593$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $24$ | $1816$ | $79908$ | $3423220$ | $147031344$ | $6321419422$ | $271818607728$ | $11688201419044$ | $502592641522044$ | $21611482588965736$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=32 x^6+24 x^5+10 x^4+25 x^3+40 x^2+14 x+18$
- $y^2=24 x^6+11 x^5+17 x^4+32 x^3+38 x^2+2 x+31$
- $y^2=22 x^6+5 x^5+10 x^4+21 x^3+41 x^2+11 x+36$
- $y^2=15 x^6+23 x^5+x^4+20 x^3+41 x^2+14 x+26$
- $y^2=8 x^6+37 x^5+28 x^4+18 x^3+26 x^2+22 x+18$
- $y^2=36 x^6+19 x^5+29 x^4+38 x^3+36 x^2+34 x+42$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{43}$.
Endomorphism algebra over $\F_{43}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-69 +20 \sqrt{3}})\). |
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 |
|---|---|---|
| 2.43.u_hb | $2$ | (not in LMFDB) |