Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 14 x + 96 x^{2} - 406 x^{3} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.0927200825158$, $\pm0.388898150193$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.1897280.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 8 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $518$ | $703444$ | $596525174$ | $499571859920$ | $420503115554998$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $16$ | $838$ | $24460$ | $706326$ | $20501196$ | $594807526$ | $17250163744$ | $500248716126$ | $14507152584640$ | $420707234701878$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=21 x^6+21 x^5+2 x^4+6 x^3+10 x^2+25 x+4$
- $y^2=11 x^6+4 x^5+21 x^3+19 x^2+13 x+17$
- $y^2=28 x^6+2 x^5+17 x^4+22 x^3+5 x^2+18 x+19$
- $y^2=21 x^6+24 x^5+13 x^4+8 x^3+16 x^2+14 x+3$
- $y^2=14 x^6+16 x^5+28 x^4+x^3+3 x^2+7 x+27$
- $y^2=27 x^6+5 x^5+8 x^4+9 x^3+18 x^2+2 x+21$
- $y^2=26 x^6+21 x^5+4 x^4+21 x^3+10 x^2+25 x+27$
- $y^2=26 x^6+15 x^5+26 x^4+17 x^3+27 x^2+8 x+8$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{29}$.
Endomorphism algebra over $\F_{29}$| The endomorphism algebra of this simple isogeny class is 4.0.1897280.1. |
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 |
|---|---|---|
| 2.29.o_ds | $2$ | (not in LMFDB) |