Invariants
Base field: | $\F_{139}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 22 x + 139 x^{2} )^{2}$ |
$1 - 44 x + 762 x^{2} - 6116 x^{3} + 19321 x^{4}$ | |
Frobenius angles: | $\pm0.117174211439$, $\pm0.117174211439$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $11$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $13924$ | $365421456$ | $7204639749316$ | $139350835364373504$ | $2692464804794951989924$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $96$ | $18910$ | $2682672$ | $373293454$ | $51889087536$ | $7212555810286$ | $1002544475411904$ | $139353668676098974$ | $19370159759770597248$ | $2692452204375011117950$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 11 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+84x^3+2$
- $y^2=47x^6+52x^5+102x^4+116x^3+9x^2+108x+16$
- $y^2=61x^6+120x^4+120x^2+61$
- $y^2=7x^6+61x^5+54x^4+71x^3+54x^2+61x+7$
- $y^2=113x^6+6x^5+56x^4+20x^3+27x^2+24x+69$
- $y^2=5x^6+44x^5+82x^4+49x^3+59x^2+54x+53$
- $y^2=58x^6+64x^5+35x^4+86x^3+35x^2+64x+58$
- $y^2=10x^6+65x^5+102x^4+11x^3+102x^2+65x+10$
- $y^2=x^6+35x^5+134x^4+134x^2+104x+1$
- $y^2=44x^6+108x^5+124x^4+128x^3+130x^2+88x+44$
- $y^2=2x^6+109x^3+61$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{139}$.
Endomorphism algebra over $\F_{139}$The isogeny class factors as 1.139.aw 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$ |
Base change
This is a primitive isogeny class.