Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 14 x + 95 x^{2} - 406 x^{3} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.0760864796921$, $\pm0.393524332268$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.96912.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $517$ | $701569$ | $595493008$ | $499287714937$ | $420454319741797$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $16$ | $836$ | $24418$ | $705924$ | $20498816$ | $594796070$ | $17250091616$ | $500248150660$ | $14507149158154$ | $420707223104036$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=27 x^6+27 x^5+23 x^4+13 x^3+x^2+17 x+3$
- $y^2=10 x^6+25 x^5+x^4+19 x^3+13 x^2+23 x+3$
- $y^2=3 x^6+9 x^5+19 x^4+28 x^3+12 x^2+11 x+22$
- $y^2=7 x^6+6 x^5+3 x^4+23 x^3+x^2+20 x+20$
- $y^2=19 x^6+22 x^5+x^4+10 x^3+13 x^2+7 x+26$
- $y^2=27 x^6+28 x^5+7 x^4+13 x^3+27 x^2+5 x+7$
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 4.0.96912.2. |
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.o_dr | $2$ | (not in LMFDB) |