Invariants
| Base field: | $\F_{43}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 18 x + 156 x^{2} - 774 x^{3} + 1849 x^{4}$ |
| Frobenius angles: | $\pm0.111629996936$, $\pm0.357330234349$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-80 -18 \sqrt{11}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $10$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1214$ | $3396772$ | $6342835574$ | $11688876696784$ | $21609336494318774$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $26$ | $1838$ | $79778$ | $3418998$ | $146993846$ | $6321300878$ | $271819421774$ | $11688212907166$ | $502592691675674$ | $21611482520782718$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=12 x^6+34 x^5+23 x^4+33 x^3+8 x^2+30 x+29$
- $y^2=36 x^6+12 x^5+16 x^4+36 x^3+5 x^2+40 x+21$
- $y^2=34 x^6+37 x^5+23 x^4+29 x^3+24 x^2+41 x+2$
- $y^2=36 x^6+14 x^5+22 x^4+26 x^3+41 x^2+36 x+5$
- $y^2=10 x^6+2 x^5+39 x^4+2 x^3+9 x^2+31 x+11$
- $y^2=23 x^6+31 x^5+39 x^3+8 x^2+11 x+37$
- $y^2=29 x^6+30 x^5+36 x^4+28 x^3+24 x^2+4 x+33$
- $y^2=30 x^6+23 x^5+6 x^4+25 x^3+4 x^2+29 x+26$
- $y^2=22 x^6+33 x^5+21 x^4+12 x^3+8 x^2+27 x+15$
- $y^2=2 x^6+41 x^5+8 x^4+11 x^3+6 x^2+6 x+19$
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 \(\Q(\sqrt{-80 -18 \sqrt{11}})\). |
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_ga | $2$ | (not in LMFDB) |