Invariants
Base field: | $\F_{131}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 41 x + 678 x^{2} - 5371 x^{3} + 17161 x^{4}$ |
Frobenius angles: | $\pm0.0540726963776$, $\pm0.201903648282$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.199121.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $12428$ | $288975856$ | $5050228210352$ | $86730657717369536$ | $1488382160842137649348$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $91$ | $16837$ | $2246452$ | $294501465$ | $38579622861$ | $5053914328582$ | $662062616572903$ | $86730203136592689$ | $11361656648498059852$ | $1488377021664440452677$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=82x^6+13x^5+121x^4+75x^3+23x^2+10x+50$
- $y^2=107x^6+106x^5+74x^4+38x^3+12x^2+118x+66$
- $y^2=121x^6+37x^5+63x^4+61x^3+58x^2+105x+113$
- $y^2=109x^6+10x^5+69x^4+65x^3+11x^2+128x+107$
- $y^2=81x^6+77x^5+113x^4+86x^3+102x^2+17x+111$
- $y^2=9x^6+88x^5+87x^4+116x^3+42x^2+71x+58$
- $y^2=86x^6+122x^5+7x^4+23x^3+29x^2+117x+101$
- $y^2=66x^6+52x^5+18x^4+60x^3+44x^2+38x+56$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{131}$.
Endomorphism algebra over $\F_{131}$The endomorphism algebra of this simple isogeny class is 4.0.199121.1. |
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 |
---|---|---|
2.131.bp_bac | $2$ | (not in LMFDB) |