Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 15 x + 109 x^{2} - 435 x^{3} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.136772039683$, $\pm0.339322847566$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-218 -30 \sqrt{21}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $501$ | $701901$ | $600332769$ | $500939022789$ | $420701095032576$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $15$ | $835$ | $24615$ | $708259$ | $20510850$ | $594816091$ | $17250036315$ | $500248649059$ | $14507159183595$ | $420707263364950$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=26 x^6+23 x^5+10 x^4+3 x^3+5 x^2+15 x+17$
- $y^2=18 x^6+7 x^5+20 x^4+6 x^3+4 x^2+23 x+11$
- $y^2=14 x^6+22 x^5+2 x^4+23 x^3+17 x^2+25 x+27$
- $y^2=21 x^6+2 x^5+6 x^3+9 x^2+28 x+18$
- $y^2=22 x^6+25 x^5+14 x^4+7 x^3+28 x^2+5 x+3$
- $y^2=27 x^6+24 x^5+24 x^4+26 x^2+23 x+1$
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 \(\Q(\sqrt{-218 -30 \sqrt{21}})\). |
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_ef | $2$ | (not in LMFDB) |