Invariants
Base field: | $\F_{83}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 17 x + 83 x^{2} )( 1 - 13 x + 83 x^{2} )$ |
$1 - 30 x + 387 x^{2} - 2490 x^{3} + 6889 x^{4}$ | |
Frobenius angles: | $\pm0.117184483028$, $\pm0.247123549255$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $6$ |
Isomorphism classes: | 10 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $4757$ | $46604329$ | $327146653424$ | $2252881645896361$ | $15516538108163352557$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $54$ | $6764$ | $572148$ | $47470740$ | $3939166794$ | $326941116518$ | $27136052904798$ | $2252292230561764$ | $186940255460442444$ | $15516041193252547964$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=62x^6+11x^5+57x^4+73x^3+57x^2+11x+62$
- $y^2=20x^6+57x^5+15x^4+33x^3+15x^2+57x+20$
- $y^2=23x^6+26x^5+33x^4+72x^3+7x^2+20x+57$
- $y^2=43x^6+59x^5+16x^4+6x^3+25x^2+27x+18$
- $y^2=18x^6+48x^5+x^4+29x^3+43x^2+38x+28$
- $y^2=52x^6+71x^5+50x^4+19x^3+50x^2+71x+52$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{83}$.
Endomorphism algebra over $\F_{83}$The isogeny class factors as 1.83.ar $\times$ 1.83.an and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
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.ae_acd | $2$ | (not in LMFDB) |
2.83.e_acd | $2$ | (not in LMFDB) |
2.83.be_ox | $2$ | (not in LMFDB) |