Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 26 x + 282 x^{2} - 1534 x^{3} + 3481 x^{4}$ |
| Frobenius angles: | $\pm0.0408324159889$, $\pm0.252883515811$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.11600.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2204$ | $11734096$ | $42143012396$ | $146841228326144$ | $511111901708972524$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $34$ | $3370$ | $205198$ | $12118254$ | $714917514$ | $42180216346$ | $2488647150646$ | $146830400878494$ | $8662995622910482$ | $511116753023853450$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=31 x^6+x^5+36 x^4+17 x^3+30 x^2+36 x+40$
- $y^2=48 x^6+15 x^5+31 x^4+22 x^3+13 x^2+33 x+34$
- $y^2=23 x^6+26 x^5+13 x^4+22 x^3+3 x+12$
- $y^2=43 x^6+55 x^5+17 x^4+8 x^3+24 x^2+53 x+42$
- $y^2=35 x^6+49 x^5+57 x^4+37 x^3+7 x^2+16 x+37$
- $y^2=x^6+49 x^5+49 x^4+26 x^3+53 x^2+33 x$
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 4.0.11600.1. |
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 |
|---|---|---|
| 2.59.ba_kw | $2$ | (not in LMFDB) |