Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + x - 16 x^{2} + 29 x^{3} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.228301058321$, $\pm0.821281154739$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.448668.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $24$ |
| Isomorphism classes: | 24 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $856$ | $681376$ | $598183072$ | $502138680448$ | $420699430626616$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $31$ | $809$ | $24526$ | $709953$ | $20510771$ | $594890246$ | $17249611487$ | $500245513249$ | $14507141016358$ | $420707175261089$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=27 x^6+6 x^5+19 x^4+23 x^2+4 x+7$
- $y^2=25 x^6+28 x^5+9 x^4+x^3+9 x^2+4 x+27$
- $y^2=13 x^6+25 x^5+25 x^4+19 x^3+25 x^2+2 x$
- $y^2=15 x^6+15 x^5+8 x^3+21 x^2+8 x+13$
- $y^2=24 x^6+16 x^4+28 x^3+27 x^2+16 x+10$
- $y^2=8 x^6+20 x^5+21 x^4+24 x^3+20 x^2+13 x+19$
- $y^2=7 x^6+18 x^5+24 x^4+22 x^3+25 x^2+3 x+17$
- $y^2=6 x^6+x^5+18 x^4+28 x^2+7 x+1$
- $y^2=22 x^6+15 x^5+3 x^3+19 x^2+18 x+22$
- $y^2=27 x^6+7 x^5+7 x^4+5 x^3+10 x^2+18 x+4$
- $y^2=15 x^6+14 x^5+9 x^4+3 x^3+22 x^2+4 x+5$
- $y^2=12 x^6+7 x^5+5 x^4+22 x^3+12 x^2+28 x+5$
- $y^2=2 x^6+16 x^5+28 x^4+12 x^3+3 x^2+21 x+10$
- $y^2=10 x^6+28 x^5+16 x^4+22 x^3+16 x^2+8 x$
- $y^2=15 x^5+17 x^4+7 x^3+15 x^2+8 x+11$
- $y^2=x^6+10 x^5+23 x^4+21 x^3+7 x^2+23 x+3$
- $y^2=18 x^6+9 x^5+4 x^4+9 x^3+18 x^2+17 x$
- $y^2=13 x^6+18 x^5+8 x^4+14 x^2+6 x+15$
- $y^2=6 x^6+17 x^5+3 x^4+5 x^3+20 x^2+21 x+19$
- $y^2=7 x^6+4 x^5+2 x^4+24 x^3+22 x^2+6 x+4$
- $y^2=3 x^6+28 x^5+11 x^4+5 x^3+24 x^2+11 x+23$
- $y^2=16 x^6+2 x^5+19 x^4+8 x^3+8 x^2+x+4$
- $y^2=18 x^6+22 x^5+5 x^4+3 x^3+9 x^2+25 x+15$
- $y^2=25 x^6+25 x^5+10 x^4+6 x^3+x^2+x+15$
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 4.0.448668.1. |
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.ab_aq | $2$ | (not in LMFDB) |