Invariants
| Base field: | $\F_{43}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 18 x + 157 x^{2} - 774 x^{3} + 1849 x^{4}$ |
| Frobenius angles: | $\pm0.122068226569$, $\pm0.353160252790$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-9 +2 \sqrt{10}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $30$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1215$ | $3400785$ | $6347155140$ | $11691167661225$ | $21610050890970375$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $26$ | $1840$ | $79832$ | $3419668$ | $146998706$ | $6321321070$ | $271819455542$ | $11688212929828$ | $502592692613816$ | $21611482510283200$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):
- $y^2=37 x^6+20 x^5+19 x^4+25 x^3+23 x^2+9 x+4$
- $y^2=32 x^6+39 x^5+22 x^4+24 x^3+3 x^2+41 x+37$
- $y^2=15 x^6+12 x^5+40 x^4+40 x^3+40 x^2+10 x+18$
- $y^2=34 x^6+24 x^5+10 x^4+4 x^3+8 x^2+27 x+36$
- $y^2=39 x^6+22 x^5+12 x^4+20 x^3+29 x^2+14 x+31$
- $y^2=28 x^6+36 x^5+16 x^4+30 x^3+22 x^2+23 x+8$
- $y^2=12 x^6+7 x^5+34 x^4+25 x^3+22 x^2+10 x+12$
- $y^2=37 x^6+13 x^5+21 x^4+6 x^3+4 x^2+3 x+42$
- $y^2=33 x^6+16 x^5+37 x^4+31 x^3+19 x^2+10 x+31$
- $y^2=32 x^6+38 x^5+26 x^4+37 x^3+13 x^2+20 x+8$
- $y^2=37 x^6+19 x^5+7 x^4+35 x^3+4 x^2+36 x+32$
- $y^2=34 x^6+36 x^5+35 x^4+28 x^3+21 x^2+13 x+32$
- $y^2=22 x^6+38 x^5+15 x^4+39 x^3+22 x^2+34 x+35$
- $y^2=8 x^6+8 x^4+20 x^3+2 x^2+18$
- $y^2=20 x^6+19 x^5+40 x^4+5 x^3+28 x^2+20 x+18$
- $y^2=28 x^6+35 x^5+32 x^4+34 x^3+41 x^2+16 x+4$
- $y^2=11 x^6+42 x^5+8 x^4+28 x^3+21 x^2+10 x+12$
- $y^2=12 x^6+16 x^5+12 x^4+10 x^3+2 x^2+6 x+3$
- $y^2=14 x^6+29 x^5+39 x^4+30 x^3+4 x^2+13 x+41$
- $y^2=12 x^6+37 x^5+31 x^4+41 x^3+8 x^2+25 x+2$
- $y^2=11 x^6+38 x^5+x^4+3 x^3+4 x^2+22 x+35$
- $y^2=27 x^6+40 x^5+41 x^4+33 x^3+40 x^2+42 x+26$
- $y^2=39 x^6+16 x^5+17 x^4+10 x^3+6 x^2+x+27$
- $y^2=2 x^6+42 x^5+13 x^4+30 x^3+x^2+38 x+32$
- $y^2=20 x^6+12 x^5+27 x^4+7 x^2+36 x+24$
- $y^2=34 x^6+20 x^4+28 x^3+35 x^2+40 x+18$
- $y^2=37 x^6+37 x^5+36 x^4+4 x^3+20 x^2+40 x+27$
- $y^2=9 x^6+23 x^5+31 x^4+8 x^3+26 x^2+38 x+30$
- $y^2=20 x^6+22 x^5+35 x^4+14 x^3+6 x^2+24 x+15$
- $y^2=16 x^6+9 x^5+6 x^4+24 x^3+17 x^2+40 x+3$
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 \(\Q(\sqrt{-9 +2 \sqrt{10}})\). |
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_gb | $2$ | (not in LMFDB) |