Invariants
Base field: | $\F_{83}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 30 x + 389 x^{2} - 2490 x^{3} + 6889 x^{4}$ |
Frobenius angles: | $\pm0.142946520854$, $\pm0.232154927040$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.590400.4 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $4759$ | $46633441$ | $327249988996$ | $2253076095172041$ | $15516782728097419039$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $54$ | $6768$ | $572328$ | $47474836$ | $3939228894$ | $326941781982$ | $27136057418538$ | $2252292232033828$ | $186940254974127624$ | $15516041184748456128$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=77x^6+50x^5+71x^4+16x^3+15x^2+10x+65$
- $y^2=73x^6+61x^5+44x^4+63x^3+70x^2+56x+25$
- $y^2=x^6+74x^5+62x^4+15x^3+64x^2+69x+18$
- $y^2=6x^6+30x^5+57x^4+77x^3+39x^2+62x+60$
- $y^2=18x^6+55x^5+7x^4+56x^3+82x^2+73x+39$
- $y^2=73x^6+x^5+67x^4+54x^3+73x^2+62x+59$
- $y^2=76x^6+64x^5+44x^4+4x^3+54x^2+66x+57$
- $y^2=53x^6+60x^5+16x^4+29x^3+27x^2+12x+58$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{83}$.
Endomorphism algebra over $\F_{83}$The endomorphism algebra of this simple isogeny class is 4.0.590400.4. |
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 |
---|---|---|
2.83.be_oz | $2$ | (not in LMFDB) |