Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 15 x + 106 x^{2} - 435 x^{3} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.0868090923144$, $\pm0.358628872989$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.866844.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $10$ |
| Isomorphism classes: | 10 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $498$ | $696204$ | $597002400$ | $499941307584$ | $420521149074378$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $15$ | $829$ | $24480$ | $706849$ | $20502075$ | $594783178$ | $17249972055$ | $500248520929$ | $14507157615840$ | $420707259865429$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=25x^6+4x^5+17x^4+27x^2+28x$
- $y^2=28x^6+7x^5+11x^4+12x^3+12x^2+3x+3$
- $y^2=5x^6+26x^5+15x^4+27x^3+8x^2+11x+4$
- $y^2=8x^6+10x^5+6x^4+23x^3+14x^2+21x+15$
- $y^2=24x^5+10x^4+x^3+17x^2+14x+24$
- $y^2=22x^6+27x^5+22x^4+21x^3+15x+8$
- $y^2=8x^6+25x^5+26x^4+22x^3+27x^2+2x+17$
- $y^2=3x^6+20x^5+2x^4+9x^3+11x^2+7x+10$
- $y^2=15x^6+19x^5+x^4+20x^3+14x^2+5x+14$
- $y^2=10x^6+20x^5+28x^3+21x^2+6x+21$
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 4.0.866844.2. |
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.p_ec | $2$ | (not in LMFDB) |