Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 18 x^{2} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.199777742710$, $\pm0.800222257290$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-10}, \sqrt{19})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $22$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $824$ | $678976$ | $594862904$ | $502170649600$ | $420707192278904$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $30$ | $806$ | $24390$ | $709998$ | $20511150$ | $594902486$ | $17249876310$ | $500245553758$ | $14507145975870$ | $420707151257606$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 22 curves (of which all are hyperelliptic):
- $y^2=24 x^6+27 x^5+12 x^4+9 x^3+14 x^2+9 x+2$
- $y^2=19 x^6+25 x^5+24 x^4+18 x^3+28 x^2+18 x+4$
- $y^2=19 x^6+21 x^5+8 x^4+19 x^3+3 x^2+23 x+9$
- $y^2=24 x^6+12 x^5+6 x^4+25 x^3+x^2+x+4$
- $y^2=20 x^6+26 x^5+x^4+20 x^3+26 x^2+25 x+15$
- $y^2=11 x^6+23 x^5+2 x^4+11 x^3+23 x^2+21 x+1$
- $y^2=18 x^6+25 x^5+21 x^4+9 x^3+5 x^2+19 x+24$
- $y^2=25 x^6+22 x^5+2 x^3+11 x$
- $y^2=21 x^6+15 x^5+4 x^3+22 x$
- $y^2=7 x^6+10 x^5+4 x^4+4 x^3+10 x^2+7 x+12$
- $y^2=14 x^6+20 x^5+8 x^4+8 x^3+20 x^2+14 x+24$
- $y^2=20 x^5+14 x^3+5 x+18$
- $y^2=11 x^5+28 x^3+10 x+7$
- $y^2=6 x^6+13 x^5+4 x^4+16 x^3+9 x^2+16 x+14$
- $y^2=12 x^6+26 x^5+8 x^4+3 x^3+18 x^2+3 x+28$
- $y^2=25 x^6+12 x^5+24 x^4+21 x^3+27 x^2+17 x+19$
- $y^2=10 x^6+20 x^5+7 x^4+28 x^3+11 x^2+4$
- $y^2=15 x^6+25 x^5+6 x^3+4 x^2+12 x+7$
- $y^2=8 x^6+2 x^5+x^4+7 x^3+5 x^2+25 x+8$
- $y^2=18 x^6+21 x^5+13 x^4+14 x^3+12 x^2+10 x+6$
- $y^2=4 x^6+28 x^5+2 x^4+7 x^3+24 x^2+9 x+10$
- $y^2=8 x^6+27 x^5+4 x^4+14 x^3+19 x^2+18 x+20$
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 \(\Q(\sqrt{-10}, \sqrt{19})\). |
| The base change of $A$ to $\F_{29^{2}}$ is 1.841.as 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-190}) \)$)$ |
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_s | $4$ | (not in LMFDB) |