Invariants
| Base field: | $\F_{43}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 20 x + 179 x^{2} - 860 x^{3} + 1849 x^{4}$ |
| Frobenius angles: | $\pm0.0853946760970$, $\pm0.310510497030$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-65 +20 \sqrt{7}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 4 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1149$ | $3342441$ | $6334027956$ | $11691327169881$ | $21610086049060989$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $24$ | $1808$ | $79668$ | $3419716$ | $146998944$ | $6321224702$ | $271818034848$ | $11688203656708$ | $502592681255724$ | $21611482855023728$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=28 x^6+34 x^5+37 x^4+34 x^3+39 x^2+7 x+17$
- $y^2=37 x^6+40 x^5+31 x^4+9 x^3+37 x^2+42 x+6$
- $y^2=33 x^6+16 x^5+24 x^4+5 x^3+17 x^2+3$
- $y^2=17 x^6+22 x^5+16 x^4+42 x^3+25 x^2+19 x+8$
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 \(\Q(\sqrt{-65 +20 \sqrt{7}})\). |
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_gx | $2$ | (not in LMFDB) |