Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 12 x + 76 x^{2} + 348 x^{3} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.552170843323$, $\pm0.899947270452$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-42 +16 \sqrt{2}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $30$ |
| Isomorphism classes: | 42 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1278$ | $713124$ | $595723086$ | $498938632848$ | $420842893996638$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $42$ | $850$ | $24426$ | $705430$ | $20517762$ | $594865474$ | $17249482194$ | $500247195358$ | $14507146310346$ | $420707271526930$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):
- $y^2=5 x^6+7 x^5+3 x^4+13 x^3+x^2+21 x+16$
- $y^2=12 x^6+23 x^5+27 x^4+9 x^3+6 x^2+26 x+20$
- $y^2=9 x^6+15 x^5+11 x^4+26 x^3+20 x^2+23 x+25$
- $y^2=23 x^6+20 x^5+14 x^4+28 x^3+4 x^2+18 x+5$
- $y^2=13 x^6+x^5+x^4+3 x^3+21 x^2+28 x+28$
- $y^2=11 x^6+13 x^5+8 x^4+16 x^3+8 x^2+19 x+3$
- $y^2=18 x^6+4 x^5+23 x^4+8 x^3+25 x^2+23 x+2$
- $y^2=11 x^6+2 x^5+12 x^4+9 x^3+10 x^2+21 x+9$
- $y^2=6 x^6+15 x^5+x^4+5 x^3+6 x^2+4 x+21$
- $y^2=25 x^6+27 x^5+6 x^4+4 x^3+3 x^2+14 x+5$
- $y^2=5 x^6+2 x^5+22 x^4+14 x^3+24 x^2+11 x+15$
- $y^2=27 x^6+7 x^5+13 x^4+8 x^3+4 x^2+18 x+19$
- $y^2=23 x^6+12 x^5+11 x^4+13 x^3+8 x^2+17 x+9$
- $y^2=16 x^6+19 x^5+14 x^4+7 x^3+11 x^2+28 x+20$
- $y^2=3 x^6+17 x^5+23 x^4+24 x^3+x^2+13 x+28$
- $y^2=4 x^6+2 x^5+5 x^4+11 x^3+26 x^2+28 x+19$
- $y^2=7 x^6+14 x^5+27 x^4+6 x^3+25 x^2+17 x+2$
- $y^2=4 x^6+9 x^4+14 x^3+23 x^2+17 x$
- $y^2=22 x^6+13 x^5+2 x^4+11 x^3+21 x^2+17 x+16$
- $y^2=22 x^6+19 x^5+26 x^4+x^3+10 x^2+14 x+19$
- $y^2=6 x^5+x^4+13 x^3+11 x^2+27 x+5$
- $y^2=26 x^6+9 x^5+25 x^4+12 x^3+9 x^2+7$
- $y^2=16 x^6+22 x^5+27 x^4+11 x^3+8 x^2+5$
- $y^2=5 x^6+26 x^5+2 x^4+3 x^3+23 x^2+8 x+9$
- $y^2=28 x^6+23 x^5+9 x^4+10 x^3+23 x^2+27 x+1$
- $y^2=21 x^6+6 x^5+12 x^4+14 x^3+27 x^2+2 x+26$
- $y^2=28 x^6+6 x^5+18 x^4+22 x^3+24 x^2+11 x+14$
- $y^2=2 x^6+11 x^5+14 x^4+10 x^3+13 x^2+5 x+10$
- $y^2=23 x^6+4 x^5+5 x^4+8 x^3+15 x^2+11 x+16$
- $y^2=6 x^6+18 x^5+5 x^3+28 x^2+3 x+15$
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 \(\Q(\sqrt{-42 +16 \sqrt{2}})\). |
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.am_cy | $2$ | (not in LMFDB) |