Invariants
Base field: | $\F_{83}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 17 x + 83 x^{2} )( 1 - 15 x + 83 x^{2} )$ |
$1 - 32 x + 421 x^{2} - 2656 x^{3} + 6889 x^{4}$ | |
Frobenius angles: | $\pm0.117184483028$, $\pm0.192168636682$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $9$ |
Isomorphism classes: | 12 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $4623$ | $46225377$ | $326758299984$ | $2252716875293049$ | $15516663756749536623$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $52$ | $6708$ | $571468$ | $47467268$ | $3939198692$ | $326942068518$ | $27136064661212$ | $2252292314118916$ | $186940255548679924$ | $15516041186322643668$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 9 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+33x^5+75x^4+66x^3+4x^2+29x+62$
- $y^2=71x^6+39x^5+16x^4+53x^3+4x^2+18x+5$
- $y^2=24x^6+47x^5+21x^3+76x+6$
- $y^2=45x^6+32x^5+51x^4+30x^3+9x^2+57x+50$
- $y^2=53x^6+49x^5+51x^4+14x^3+x^2+64x+8$
- $y^2=74x^6+40x^5+35x^4+34x^3+39x^2+29x+35$
- $y^2=10x^6+73x^5+9x^4+78x^3+63x^2+8x+27$
- $y^2=44x^6+62x^5+10x^4+68x^3+64x^2+13x+81$
- $y^2=17x^6+42x^5+38x^4+66x^3+63x^2+79x+41$
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.ap 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.ac_adl | $2$ | (not in LMFDB) |
2.83.c_adl | $2$ | (not in LMFDB) |
2.83.bg_qf | $2$ | (not in LMFDB) |