Invariants
| Base field: | $\F_{43}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 18 x + 165 x^{2} - 774 x^{3} + 1849 x^{4}$ |
| Frobenius angles: | $\pm0.207954997452$, $\pm0.303673090206$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-89 -18 \sqrt{2}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $10$ |
| Isomorphism classes: | 10 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1223$ | $3432961$ | $6381746084$ | $11709009493321$ | $21614814079014623$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $26$ | $1856$ | $80264$ | $3424884$ | $147031106$ | $6321341486$ | $271817793350$ | $11688195410788$ | $502592600671688$ | $21611482320462896$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=20 x^6+40 x^5+30 x^4+24 x^3+4 x^2+x+29$
- $y^2=22 x^6+25 x^5+11 x^4+40 x^3+20 x^2+14 x+2$
- $y^2=18 x^6+5 x^5+36 x^4+8 x^3+9 x^2+15 x+34$
- $y^2=33 x^6+42 x^5+33 x^4+41 x^3+25 x^2+7 x+17$
- $y^2=29 x^6+30 x^5+14 x^4+35 x^3+31 x^2+14 x+18$
- $y^2=33 x^6+41 x^5+40 x^4+7 x^3+34 x^2+28 x+20$
- $y^2=13 x^6+14 x^5+26 x^4+4 x^3+10 x^2+33 x+28$
- $y^2=34 x^6+5 x^5+13 x^4+28 x^3+x^2+33 x+8$
- $y^2=26 x^6+36 x^5+29 x^4+5 x^3+23 x^2+11 x+25$
- $y^2=12 x^6+31 x^5+23 x^4+18 x^3+25 x^2+39$
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 \(\Q(\sqrt{-89 -18 \sqrt{2}})\). |
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 |
|---|---|---|
| 2.43.s_gj | $2$ | (not in LMFDB) |