Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 26 x + 284 x^{2} - 1534 x^{3} + 3481 x^{4}$ |
| Frobenius angles: | $\pm0.0914926884064$, $\pm0.237894336644$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.297792.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 8 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2206$ | $11749156$ | $42175272022$ | $146879347929808$ | $511143316712449366$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $34$ | $3374$ | $205354$ | $12121398$ | $714961454$ | $42180689486$ | $2488651266758$ | $146830430619934$ | $8662995808722082$ | $511116754148953214$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=23 x^6+4 x^5+2 x^4+9 x^3+16 x^2+49 x+28$
- $y^2=21 x^6+44 x^5+32 x^4+41 x^3+51 x^2+19 x+23$
- $y^2=10 x^6+9 x^5+47 x^4+54 x^3+23 x^2+46 x+42$
- $y^2=36 x^6+54 x^5+43 x^4+6 x^3+52 x^2+56 x+28$
- $y^2=44 x^6+48 x^5+3 x^3+21 x^2+33 x+42$
- $y^2=23 x^6+10 x^5+20 x^4+40 x^3+40 x^2+52 x+2$
- $y^2=3 x^6+23 x^5+10 x^4+21 x^3+16 x^2+5 x+28$
- $y^2=39 x^6+4 x^5+48 x^4+32 x^3+24 x^2+15 x+23$
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 4.0.297792.1. |
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_ky | $2$ | (not in LMFDB) |