Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 47 x^{2} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.0996399008450$, $\pm0.900360099155$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-11}, \sqrt{105})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $12$ |
| Isomorphism classes: | 32 |
| Cyclic group of points: | yes |
This isogeny class is simple but not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $795$ | $632025$ | $594838080$ | $499502630025$ | $420707274319875$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $30$ | $748$ | $24390$ | $706228$ | $20511150$ | $594852838$ | $17249876310$ | $500248686628$ | $14507145975870$ | $420707315339548$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=19 x^6+22 x^5+12 x^4+21 x^3+23 x^2+13 x+4$
- $y^2=5 x^6+11 x^5+25 x^4+4 x^3+15 x^2+4 x+22$
- $y^2=10 x^6+22 x^5+21 x^4+8 x^3+x^2+8 x+15$
- $y^2=12 x^6+17 x^5+20 x^4+22 x^3+12 x^2+12 x+16$
- $y^2=24 x^6+5 x^5+11 x^4+15 x^3+24 x^2+24 x+3$
- $y^2=14 x^6+20 x^5+13 x^4+28 x^3+15 x^2+18 x+15$
- $y^2=28 x^6+11 x^5+26 x^4+27 x^3+x^2+7 x+1$
- $y^2=24 x^6+18 x^5+6 x^4+11 x^3+15 x^2+25 x+15$
- $y^2=3 x^6+16 x^5+9 x^4+6 x^3+11 x^2+16 x+23$
- $y^2=11 x^6+8 x^5+12 x^4+16 x^3+2 x^2+10 x+18$
- $y^2=2 x^6+12 x^5+17 x^4+18 x^3+20 x^2+x+14$
- $y^2=4 x^6+24 x^5+5 x^4+7 x^3+11 x^2+2 x+28$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{29^{2}}$.
Endomorphism algebra over $\F_{29}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-11}, \sqrt{105})\). |
| The base change of $A$ to $\F_{29^{2}}$ is 1.841.abv 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1155}) \)$)$ |
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.a_bv | $4$ | (not in LMFDB) |