Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 61 x^{2} + 3481 x^{4}$ |
| Frobenius angles: | $\pm0.163533810356$, $\pm0.836466189644$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-57}, \sqrt{179})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $24$ |
| Cyclic group of points: | yes |
This isogeny class is simple but not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3421$ | $11703241$ | $42180943684$ | $146909017083609$ | $511116752710854301$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $60$ | $3360$ | $205380$ | $12123844$ | $714924300$ | $42181353726$ | $2488651484820$ | $146830465065604$ | $8662995818654940$ | $511116752121067200$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=24 x^6+54 x^5+41 x^4+52 x^3+31 x^2+14 x+25$
- $y^2=48 x^6+49 x^5+23 x^4+45 x^3+3 x^2+28 x+50$
- $y^2=38 x^6+22 x^5+13 x^4+50 x^3+13 x^2+7 x+40$
- $y^2=17 x^6+44 x^5+26 x^4+41 x^3+26 x^2+14 x+21$
- $y^2=51 x^6+27 x^5+29 x^4+21 x^3+38 x^2+18 x+28$
- $y^2=43 x^6+54 x^5+58 x^4+42 x^3+17 x^2+36 x+56$
- $y^2=15 x^6+40 x^5+48 x^4+8 x^3+23 x^2+52 x+5$
- $y^2=11 x^6+49 x^5+34 x^4+13 x^3+24 x^2+34 x+34$
- $y^2=22 x^6+39 x^5+9 x^4+26 x^3+48 x^2+9 x+9$
- $y^2=52 x^6+57 x^5+45 x^4+58 x^3+8 x^2+36 x+36$
- $y^2=29 x^6+23 x^5+53 x^4+44 x^3+11 x^2+21 x+7$
- $y^2=58 x^6+46 x^5+47 x^4+29 x^3+22 x^2+42 x+14$
- $y^2=44 x^6+4 x^5+27 x^4+26 x^3+6 x^2+12 x+17$
- $y^2=14 x^6+16 x^5+42 x^4+15 x^3+42 x^2+46 x+54$
- $y^2=28 x^6+37 x^5+28 x^4+16 x^3+25 x^2+20 x+26$
- $y^2=9 x^6+51 x^5+28 x^4+18 x^3+18 x^2+51 x+27$
- $y^2=18 x^6+43 x^5+56 x^4+36 x^3+36 x^2+43 x+54$
- $y^2=57 x^6+5 x^5+31 x^4+35 x^3+41 x^2+3 x+4$
- $y^2=55 x^6+10 x^5+3 x^4+11 x^3+23 x^2+6 x+8$
- $y^2=x^6+44 x^5+19 x^4+47 x^3+9 x^2+11 x+43$
- $y^2=11 x^6+43 x^5+44 x^4+40 x^3+29 x^2+57 x+32$
- $y^2=41 x^6+27 x^5+10 x^4+17 x^3+35 x^2+21 x+52$
- $y^2=23 x^6+54 x^5+20 x^4+34 x^3+11 x^2+42 x+45$
- $y^2=13 x^6+17 x^5+37 x^4+4 x^3+52 x^2+57 x+11$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{59^{2}}$.
Endomorphism algebra over $\F_{59}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-57}, \sqrt{179})\). |
| The base change of $A$ to $\F_{59^{2}}$ is 1.3481.acj 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10203}) \)$)$ |
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.a_cj | $4$ | (not in LMFDB) |