Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 41 x^{2} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.125047311222$, $\pm0.874952688778$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{11}, \sqrt{-17})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $15$ |
| Isomorphism classes: | 100 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $801$ | $641601$ | $594857844$ | $500249242089$ | $420707262264201$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $30$ | $760$ | $24390$ | $707284$ | $20511150$ | $594892366$ | $17249876310$ | $500249242084$ | $14507145975870$ | $420707291228200$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 15 curves (of which all are hyperelliptic):
- $y^2=9 x^6+x^5+26 x^4+25 x^3+6 x^2+23 x+17$
- $y^2=18 x^6+2 x^5+23 x^4+21 x^3+12 x^2+17 x+5$
- $y^2=27 x^6+23 x^5+23 x^4+23 x^3+13 x^2+18 x+16$
- $y^2=25 x^6+17 x^5+17 x^4+17 x^3+26 x^2+7 x+3$
- $y^2=7 x^6+19 x^5+20 x^4+13 x^3+8 x^2+10 x+3$
- $y^2=27 x^6+12 x^5+22 x^4+15 x^3+19 x^2+2 x+23$
- $y^2=x^6+13 x^5+23 x^4+25 x^3+8 x^2+24 x+6$
- $y^2=2 x^6+26 x^5+17 x^4+21 x^3+16 x^2+19 x+12$
- $y^2=x^6+3 x^5+21 x^4+6 x^3+18 x^2+28 x+3$
- $y^2=2 x^6+6 x^5+13 x^4+12 x^3+7 x^2+27 x+6$
- $y^2=10 x^6+5 x^5+20 x^4+23 x^3+15 x^2+x+16$
- $y^2=15 x^6+4 x^4+9 x^3+10 x^2+6$
- $y^2=4 x^6+15 x^5+11 x^4+8 x^3+8 x^2+12 x+22$
- $y^2=8 x^6+x^5+22 x^4+16 x^3+16 x^2+24 x+15$
- $y^2=27 x^6+7 x^5+21 x^4+2 x^3+7 x^2+4 x+1$
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 \(\Q(\sqrt{11}, \sqrt{-17})\). |
| The base change of $A$ to $\F_{29^{2}}$ is 1.841.abp 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-187}) \)$)$ |
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_bp | $4$ | (not in LMFDB) |