Invariants
| Base field: | $\F_{43}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 21 x + 192 x^{2} - 903 x^{3} + 1849 x^{4}$ |
| Frobenius angles: | $\pm0.0927922568847$, $\pm0.277512598900$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-230 +42 \sqrt{17}})\) |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1118$ | $3315988$ | $6331363688$ | $11695118285344$ | $21612313035019538$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $23$ | $1793$ | $79634$ | $3420825$ | $147014093$ | $6321312434$ | $271818057359$ | $11688199727665$ | $502592650556726$ | $21611482823443313$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=5 x^6+15 x^5+14 x^3+22 x^2+2 x+35$
- $y^2=4 x^6+29 x^5+14 x^4+5 x^3+33 x^2+22 x+39$
- $y^2=17 x^6+30 x^5+6 x^4+22 x^3+12 x^2+32 x+1$
- $y^2=28 x^6+2 x^5+25 x^4+5 x^2+38 x+12$
- $y^2=33 x^6+10 x^5+4 x^4+25 x^3+30 x^2+7 x+28$
- $y^2=20 x^6+8 x^5+41 x^4+12 x^3+41 x^2+35 x+42$
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 \(\Q(\sqrt{-230 +42 \sqrt{17}})\). |
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.v_hk | $2$ | (not in LMFDB) |