Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 13 x + 85 x^{2} - 377 x^{3} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.0831292265081$, $\pm0.422548196387$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.3144245.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $537$ | $707229$ | $594508941$ | $498981884805$ | $420484379299152$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $17$ | $843$ | $24377$ | $705491$ | $20500282$ | $594828531$ | $17250203533$ | $500247633091$ | $14507145127373$ | $420707237541078$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+14x^5+27x^4+10x^3+12x^2+16x+19$
- $y^2=9x^6+7x^5+x^4+2x^3+25x^2+20x+26$
- $y^2=5x^6+5x^5+22x^4+2x^3+5x^2+26x+5$
- $y^2=12x^6+18x^5+27x^4+21x^3+15x^2+2x+2$
- $y^2=23x^6+8x^5+9x^4+6x^3+27x+6$
- $y^2=18x^6+14x^5+18x^4+21x^3+10x^2+17x+18$
- $y^2=17x^6+20x^5+12x^4+14x^3+4x^2+26x+20$
- $y^2=12x^6+14x^5+6x^4+24x^3+24x^2+13x+15$
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 4.0.3144245.1. |
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.n_dh | $2$ | (not in LMFDB) |