Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 23 x + 243 x^{2} - 1357 x^{3} + 3481 x^{4}$ |
| Frobenius angles: | $\pm0.125017556913$, $\pm0.305657502371$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-386 -46 \sqrt{29}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $13$ |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2345$ | $11971225$ | $42289286795$ | $146895035269325$ | $511128001822387600$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $37$ | $3439$ | $205909$ | $12122691$ | $714940032$ | $42180468067$ | $2488651543723$ | $146830457691891$ | $8662996122607231$ | $511116755718462774$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 13 curves (of which all are hyperelliptic):
- $y^2=18 x^6+23 x^5+48 x^4+6 x^3+56 x^2+14 x+50$
- $y^2=46 x^6+39 x^5+16 x^4+16 x^3+47 x^2+41 x+2$
- $y^2=50 x^6+55 x^5+58 x^4+5 x^3+32 x^2+53 x+42$
- $y^2=32 x^6+14 x^5+24 x^4+5 x^3+56 x^2+22 x+38$
- $y^2=16 x^6+7 x^5+36 x^4+36 x^3+35 x^2+52 x$
- $y^2=12 x^6+57 x^5+46 x^4+5 x^3+46 x^2+24 x+32$
- $y^2=29 x^6+57 x^5+16 x^4+5 x^3+51 x^2+6 x+52$
- $y^2=34 x^6+x^5+x^4+48 x^3+9 x^2+11 x+53$
- $y^2=6 x^6+23 x^5+x^4+18 x^3+55 x^2+43 x+30$
- $y^2=55 x^6+43 x^5+10 x^4+29 x^3+52 x^2+52 x+18$
- $y^2=21 x^6+8 x^5+13 x^4+18 x^3+x^2+45 x+50$
- $y^2=24 x^6+43 x^5+16 x^4+43 x^3+27 x^2+41 x+14$
- $y^2=19 x^6+11 x^5+41 x^4+58 x^3+34 x^2+16 x+39$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{59}$.
Endomorphism algebra over $\F_{59}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-386 -46 \sqrt{29}})\). |
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 |
|---|---|---|
| 2.59.x_jj | $2$ | (not in LMFDB) |