Invariants
Base field: | $\F_{163}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 22 x + 163 x^{2} )^{2}$ |
$1 - 44 x + 810 x^{2} - 7172 x^{3} + 26569 x^{4}$ | |
Frobenius angles: | $\pm0.169471200781$, $\pm0.169471200781$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $32$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $20164$ | $697593744$ | $18756331016164$ | $498351193239684096$ | $13239774480732511599364$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $120$ | $26254$ | $4330968$ | $705968110$ | $115064820840$ | $18755386876798$ | $3057125425570056$ | $498311415554219614$ | $81224760530998111704$ | $13239635966753852127214$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 32 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=116x^6+154x^5+28x^4+57x^3+28x^2+154x+116$
- $y^2=36x^6+135x^5+117x^4+122x^3+16x^2+20x+114$
- $y^2=129x^6+15x^4+15x^2+129$
- $y^2=63x^6+78x^5+28x^4+160x^3+2x^2+32x+52$
- $y^2=70x^6+49x^5+95x^4+144x^3+7x^2+67x+127$
- $y^2=115x^6+79x^5+89x^4+77x^3+89x^2+79x+115$
- $y^2=68x^6+71x^5+157x^4+82x^3+23x^2+116x+39$
- $y^2=47x^6+57x^5+145x^4+67x^3+55x^2+96x+46$
- $y^2=80x^6+97x^4+97x^2+80$
- $y^2=156x^6+71x^5+41x^4+7x^3+41x^2+71x+156$
- $y^2=97x^6+132x^5+13x^4+34x^3+13x^2+132x+97$
- $y^2=33x^6+25x^5+9x^4+103x^3+9x^2+25x+33$
- $y^2=127x^6+38x^5+50x^4+43x^3+45x^2+121x+10$
- $y^2=117x^6+112x^5+48x^4+81x^3+106x^2+58x+98$
- $y^2=114x^6+130x^5+94x^4+30x^3+130x^2+54x+52$
- $y^2=55x^6+155x^5+105x^4+36x^3+105x^2+155x+55$
- $y^2=71x^6+74x^5+61x^4+89x^3+150x^2+72x+32$
- $y^2=75x^6+29x^5+32x^4+x^3+162x^2+126x+24$
- $y^2=2x^6+66x^3+154$
- $y^2=90x^6+45x^5+13x^4+16x^3+92x^2+162x+119$
- and 12 more
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{163}$.
Endomorphism algebra over $\F_{163}$The isogeny class factors as 1.163.aw 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-42}) \)$)$ |
Base change
This is a primitive isogeny class.