Invariants
| Base field: | $\F_{167}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 23 x + 167 x^{2} )^{2}$ |
| $1 - 46 x + 863 x^{2} - 7682 x^{3} + 27889 x^{4}$ | |
| Frobenius angles: | $\pm0.150776270497$, $\pm0.150776270497$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $21$ |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $21025$ | $767013025$ | $21685972512400$ | $604994735268105625$ | $16872061939432877175625$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $122$ | $27500$ | $4656176$ | $777831828$ | $129893017342$ | $21691979396750$ | $3622557823702786$ | $604967119441982308$ | $101029508549971808912$ | $16871927924916425577500$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 21 curves (of which all are hyperelliptic):
- $y^2=46 x^6+160 x^5+163 x^4+161 x^3+163 x^2+160 x+46$
- $y^2=37 x^6+11 x^5+130 x^4+147 x^3+130 x^2+11 x+37$
- $y^2=103 x^6+89 x^5+112 x^4+145 x^3+91 x^2+56 x+101$
- $y^2=4 x^6+156 x^5+26 x^4+103 x^3+26 x^2+156 x+4$
- $y^2=142 x^6+63 x^5+42 x^4+128 x^3+72 x^2+166 x+35$
- $y^2=50 x^6+11 x^5+83 x^4+149 x^3+136 x^2+2 x+26$
- $y^2=13 x^6+31 x^5+84 x^4+34 x^3+84 x^2+31 x+13$
- $y^2=37 x^6+82 x^5+125 x^4+120 x^3+125 x^2+82 x+37$
- $y^2=35 x^6+37 x^5+126 x^4+92 x^3+126 x^2+37 x+35$
- $y^2=21 x^6+55 x^5+139 x^4+79 x^3+139 x^2+55 x+21$
- $y^2=131 x^6+152 x^5+63 x^4+41 x^3+63 x^2+152 x+131$
- $y^2=44 x^6+125 x^5+130 x^4+120 x^3+146 x^2+69 x+156$
- $y^2=146 x^6+53 x^5+21 x^4+72 x^3+21 x^2+53 x+146$
- $y^2=13 x^6+14 x^5+71 x^4+x^3+59 x^2+151 x+34$
- $y^2=86 x^6+14 x^5+16 x^4+47 x^3+16 x^2+14 x+86$
- $y^2=143 x^6+64 x^5+142 x^4+98 x^3+142 x^2+64 x+143$
- $y^2=84 x^6+98 x^5+24 x^4+30 x^3+123 x^2+99 x+91$
- $y^2=135 x^6+33 x^5+6 x^4+114 x^3+6 x^2+33 x+135$
- $y^2=68 x^6+61 x^5+16 x^4+124 x^3+53 x^2+166 x+107$
- $y^2=51 x^6+60 x^5+25 x^4+25 x^3+78 x^2+152 x+98$
- $y^2=20 x^6+159 x^5+104 x^4+38 x^3+81 x^2+18 x+139$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{167}$.
Endomorphism algebra over $\F_{167}$| The isogeny class factors as 1.167.ax 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-139}) \)$)$ |
Base change
This is a primitive isogeny class.