Invariants
Base field: | $\F_{181}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 49 x + 961 x^{2} - 8869 x^{3} + 32761 x^{4}$ |
Frobenius angles: | $\pm0.0989332971490$, $\pm0.164773451880$ |
Angle rank: | $2$ (numerical) |
Number field: | \(\Q(\zeta_{5})\) |
Galois group: | $C_4$ |
Jacobians: | $11$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $24805$ | $1057710005$ | $35144116439905$ | $1151947736663190005$ | $37738739669628951250000$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $133$ | $32283$ | $5926753$ | $1073293443$ | $194264980098$ | $35161843616283$ | $6364291159308733$ | $1151936660686152003$ | $208500535094782804993$ | $37738596847170094900198$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 11 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=8x^5+180$
- $y^2=171x^6+104x^5+51x^4+45x^3+23x^2+2x+16$
- $y^2=121x^6+92x^5+64x^4+124x^3+52x^2+53x+48$
- $y^2=106x^6+155x^5+91x^4+9x^3+163x^2+103x+86$
- $y^2=57x^6+130x^5+108x^4+127x^3+39x^2+56x+17$
- $y^2=142x^5+37x^4+145x^3+163x^2+20x+111$
- $y^2=59x^6+75x^5+126x^4+38x^3+143x^2+23x+151$
- $y^2=105x^6+59x^5+32x^4+60x^3+116x^2+107x+107$
- $y^2=92x^6+55x^5+164x^4+91x^3+69x^2+33x+130$
- $y^2=28x^6+3x^5+56x^4+128x^3+17x^2+116x+178$
- $y^2=28x^5+53x^4+60x^3+80x^2+173x+83$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{181}$.
Endomorphism algebra over $\F_{181}$The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{5})\). |
Base change
This is a primitive isogeny class.