Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 4 x - 6 x^{2} + 116 x^{3} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.303073580111$, $\pm0.900289149367$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.245072.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $28$ |
| Isomorphism classes: | 40 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $956$ | $684496$ | $606732092$ | $500810129408$ | $420699151889116$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $34$ | $814$ | $24874$ | $708078$ | $20510754$ | $594797086$ | $17249479498$ | $500247219294$ | $14507143746082$ | $420707315152014$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):
- $y^2=x^6+12 x^5+26 x^4+26 x^3+19 x^2+7 x+26$
- $y^2=28 x^6+16 x^5+15 x^4+17 x^3+18 x^2+3 x+18$
- $y^2=20 x^6+11 x^5+2 x^4+10 x^3+26 x^2+14 x+9$
- $y^2=13 x^6+15 x^5+16 x^4+15 x^3+15 x^2+3 x+1$
- $y^2=7 x^6+9 x^5+26 x^4+20 x^3+8 x^2+17 x+4$
- $y^2=12 x^6+22 x^4+24 x^3+7 x^2+14 x$
- $y^2=16 x^6+9 x^5+15 x^4+5 x^3+26 x^2+7 x+25$
- $y^2=12 x^6+18 x^5+15 x^4+19 x^3+15 x^2+11 x+17$
- $y^2=22 x^6+7 x^5+13 x^4+23 x^3+10 x^2+26 x+22$
- $y^2=27 x^6+28 x^5+4 x^4+6 x^3+12 x^2+27 x+4$
- $y^2=15 x^6+11 x^5+14 x^4+16 x^3+19 x^2+2 x$
- $y^2=3 x^6+26 x^5+x^4+21 x^3+2 x^2+12 x+2$
- $y^2=7 x^6+21 x^5+3 x^3+24 x^2+27 x+26$
- $y^2=20 x^6+20 x^5+27 x^4+15 x^3+24 x^2+28 x+10$
- $y^2=15 x^6+3 x^5+19 x^4+20 x^2+19 x+18$
- $y^2=2 x^5+14 x^4+17 x^3+3 x^2+8 x+6$
- $y^2=14 x^6+10 x^4+12 x^3+4 x^2+11 x+4$
- $y^2=21 x^6+15 x^5+27 x^3+25 x^2+12 x+11$
- $y^2=8 x^6+21 x^5+22 x^4+22 x^3+12 x^2+x+2$
- $y^2=19 x^6+12 x^5+21 x^4+5 x^3+23 x^2+28 x+2$
- $y^2=5 x^6+2 x^5+4 x^4+7 x^3+2 x^2+20 x+28$
- $y^2=15 x^6+24 x^5+15 x^4+19 x^3+10 x^2+21 x+6$
- $y^2=12 x^6+25 x^5+3 x^4+17 x^3+21 x^2+16 x+17$
- $y^2=6 x^6+3 x^5+12 x^4+x^3+2 x+24$
- $y^2=12 x^6+26 x^5+3 x^4+24 x^3+x^2+x+3$
- $y^2=11 x^6+5 x^5+9 x^4+12 x^3+13 x^2+2 x+26$
- $y^2=14 x^6+13 x^5+26 x^4+16 x^2+3 x+25$
- $y^2=4 x^6+2 x^5+26 x^4+25 x^3+27 x^2+8 x+24$
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 4.0.245072.1. |
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.ae_ag | $2$ | (not in LMFDB) |