Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 16 x + 117 x^{2} - 464 x^{3} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.100679126603$, $\pm0.320248433638$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-122 +2 \sqrt{5}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 6 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $479$ | $689281$ | $597937616$ | $500644779449$ | $420651155542399$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $14$ | $820$ | $24518$ | $707844$ | $20508414$ | $594791686$ | $17249843366$ | $500247845124$ | $14507160366782$ | $420707307311700$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=27 x^6+7 x^5+2 x^4+19 x^3+22 x^2+16 x+26$
- $y^2=12 x^6+13 x^5+x^4+9 x^3+19 x+7$
- $y^2=12 x^6+21 x^4+6 x^3+10 x^2+5 x+2$
- $y^2=2 x^6+26 x^4+3 x^3+15 x^2+26 x+13$
- $y^2=16 x^6+11 x^5+11 x^3+27$
- $y^2=12 x^6+3 x^5+21 x^4+6 x^3+5 x^2+x+18$
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 \(\Q(\sqrt{-122 +2 \sqrt{5}})\). |
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.q_en | $2$ | (not in LMFDB) |