Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 5 x + 56 x^{2} + 145 x^{3} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.488995282460$, $\pm0.666227174749$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-406 +10 \sqrt{33}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $30$ |
| Isomorphism classes: | 30 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1048$ | $783904$ | $588032800$ | $499657273984$ | $420769313411128$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $35$ | $929$ | $24110$ | $706449$ | $20514175$ | $594822278$ | $17250068395$ | $500245772929$ | $14507136026630$ | $420707291913929$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):
- $y^2=7 x^6+8 x^5+12 x^4+16 x^3+5 x^2+9 x+12$
- $y^2=x^5+11 x^4+5 x^3+15 x^2+26 x$
- $y^2=28 x^6+24 x^5+6 x^4+20 x^3+15 x^2+2 x+26$
- $y^2=26 x^6+x^5+16 x^4+2 x^3+13 x^2+27 x+20$
- $y^2=23 x^6+2 x^5+8 x^4+18 x^3+x^2+8 x+15$
- $y^2=4 x^6+2 x^5+12 x^4+14 x^2+x+6$
- $y^2=9 x^6+6 x^5+9 x^4+10 x^3+15 x^2+24 x+2$
- $y^2=5 x^6+22 x^5+8 x^4+15 x^3+27 x^2+26 x+1$
- $y^2=7 x^6+x^5+26 x^3+12 x^2+11 x+12$
- $y^2=24 x^6+x^5+12 x^4+5 x^3+10 x^2+20 x+4$
- $y^2=5 x^6+28 x^5+3 x^4+13 x^3+23 x^2+17 x+4$
- $y^2=25 x^6+11 x^5+25 x^4+11 x^3+20 x^2+4 x+24$
- $y^2=24 x^5+5 x^3+7 x^2+15 x+25$
- $y^2=21 x^6+15 x^5+21 x^4+13 x^3+19 x^2+27 x$
- $y^2=22 x^5+23 x^4+28 x^3+8 x^2+3 x+22$
- $y^2=17 x^6+16 x^5+9 x^4+x^3+17 x^2+2$
- $y^2=4 x^6+16 x^5+8 x^4+19 x^3+18 x^2+15 x+9$
- $y^2=6 x^6+9 x^5+27 x^4+5 x^3+28 x^2+27 x+28$
- $y^2=27 x^6+7 x^5+13 x^4+19 x^3+8 x^2+13 x+23$
- $y^2=12 x^6+7 x^5+8 x^4+26 x^3+4 x^2+17 x+5$
- $y^2=12 x^6+10 x^5+16 x^4+16 x^3+x^2+14 x+6$
- $y^2=3 x^6+15 x^5+22 x^4+10 x^2+x$
- $y^2=7 x^6+20 x^5+21 x^4+2 x^3+5 x^2+2 x+24$
- $y^2=20 x^6+10 x^5+26 x^4+17 x^3+18 x^2+4 x+21$
- $y^2=17 x^6+21 x^4+10 x^3+4 x^2+4 x+21$
- $y^2=21 x^6+13 x^5+25 x^4+23 x^3+7 x^2+7 x+26$
- $y^2=12 x^6+4 x^5+19 x^4+21 x^3+21 x^2+8 x+9$
- $y^2=28 x^6+22 x^5+3 x^4+6 x^3+22 x^2+2 x+26$
- $y^2=3 x^6+8 x^5+6 x^4+6 x^3+6 x^2+25 x$
- $y^2=15 x^6+14 x^5+27 x^4+24 x^3+28 x^2+18 x+24$
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 \(\Q(\sqrt{-406 +10 \sqrt{33}})\). |
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.af_ce | $2$ | (not in LMFDB) |