Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 25 x + 273 x^{2} - 1475 x^{3} + 3481 x^{4}$ |
| Frobenius angles: | $\pm0.153158091399$, $\pm0.234397866783$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.134525.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $7$ |
| Isomorphism classes: | 7 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2255$ | $11850025$ | $42268168745$ | $146942454854525$ | $511177882576042000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $35$ | $3403$ | $205805$ | $12126603$ | $715009800$ | $42181050223$ | $2488653208895$ | $146830433624163$ | $8662995712809965$ | $511116752485892598$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which all are hyperelliptic):
- $y^2=21 x^6+57 x^5+45 x^4+38 x^3+50 x^2+9 x+47$
- $y^2=6 x^6+4 x^5+48 x^4+49 x^3+45 x^2+19 x+29$
- $y^2=8 x^6+38 x^5+8 x^4+2 x^3+26 x^2+50 x+9$
- $y^2=44 x^6+8 x^5+39 x^4+3 x^3+28 x^2+8 x+44$
- $y^2=53 x^6+56 x^5+44 x^4+41 x^3+49 x^2+37 x+11$
- $y^2=36 x^6+28 x^5+35 x^4+19 x^3+33 x^2+57 x+24$
- $y^2=12 x^6+29 x^5+57 x^4+43 x^3+16 x^2+27 x+45$
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 4.0.134525.1. |
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.z_kn | $2$ | (not in LMFDB) |