Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 3 x^{2} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.258235827353$, $\pm0.741764172647$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{55}, \sqrt{-61})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $20$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $845$ | $714025$ | $594815780$ | $502617192025$ | $420707243796125$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $30$ | $848$ | $24390$ | $710628$ | $20511150$ | $594808238$ | $17249876310$ | $500243644228$ | $14507145975870$ | $420707254292048$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):
- $y^2=15 x^6+19 x^5+13 x^4+2 x^3+21 x^2+12 x+20$
- $y^2=11 x^6+12 x^5+24 x^4+21 x^3+x^2+7 x+20$
- $y^2=22 x^6+24 x^5+19 x^4+13 x^3+2 x^2+14 x+11$
- $y^2=23 x^6+7 x^5+4 x^3+11 x^2+7 x+23$
- $y^2=17 x^6+14 x^5+8 x^3+22 x^2+14 x+17$
- $y^2=16 x^6+7 x^4+18 x^3+14 x^2+3 x+27$
- $y^2=3 x^6+14 x^4+7 x^3+28 x^2+6 x+25$
- $y^2=28 x^6+24 x^5+8 x^4+8 x^3+7 x^2+22 x+14$
- $y^2=3 x^6+19 x^5+13 x^4+11 x^2+12 x+16$
- $y^2=6 x^6+9 x^5+26 x^4+22 x^2+24 x+3$
- $y^2=17 x^6+20 x^5+26 x^4+x^2+3 x+9$
- $y^2=5 x^6+11 x^5+23 x^4+2 x^2+6 x+18$
- $y^2=28 x^6+12 x^5+15 x^4+4 x^3+20 x^2+2 x+3$
- $y^2=21 x^6+6 x^5+26 x^4+x^3+8 x^2+23 x+11$
- $y^2=13 x^6+12 x^5+23 x^4+2 x^3+16 x^2+17 x+22$
- $y^2=14 x^6+17 x^5+24 x^4+5 x^3+11 x^2+15 x+24$
- $y^2=16 x^6+13 x^5+19 x^4+13 x^3+5 x^2+13 x+21$
- $y^2=3 x^6+26 x^5+9 x^4+26 x^3+10 x^2+26 x+13$
- $y^2=12 x^6+2 x^5+8 x^4+9 x^3+16 x^2+28 x+10$
- $y^2=24 x^6+4 x^5+16 x^4+18 x^3+3 x^2+27 x+20$
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 \(\Q(\sqrt{55}, \sqrt{-61})\). |
| The base change of $A$ to $\F_{29^{2}}$ is 1.841.d 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3355}) \)$)$ |
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_ad | $4$ | (not in LMFDB) |