Invariants
| Base field: | $\F_{83}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 28 x + 348 x^{2} - 2324 x^{3} + 6889 x^{4}$ |
| Frobenius angles: | $\pm0.0731644126318$, $\pm0.309647214466$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.12255488.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $24$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4886$ | $46856740$ | $327115906838$ | $2252374440621200$ | $15515764249987534246$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $56$ | $6802$ | $572096$ | $47460054$ | $3938970336$ | $326939124034$ | $27136042378760$ | $2252292250476126$ | $186940256358455768$ | $15516041199965196882$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=18 x^6+57 x^5+30 x^4+6 x^3+2 x^2+66 x+1$
- $y^2=8 x^6+77 x^5+33 x^4+64 x^2+61 x+72$
- $y^2=36 x^6+24 x^5+50 x^4+6 x^3+11 x^2+28 x+54$
- $y^2=46 x^6+74 x^5+52 x^4+41 x^3+74 x^2+42 x+10$
- $y^2=81 x^6+66 x^5+2 x^4+52 x^3+5 x^2+29 x+42$
- $y^2=74 x^6+3 x^5+6 x^4+15 x^3+22 x^2+43 x+69$
- $y^2=67 x^6+29 x^5+73 x^4+42 x^3+69 x^2+27 x+72$
- $y^2=79 x^6+13 x^5+53 x^4+2 x^3+75 x^2+27 x+24$
- $y^2=22 x^6+81 x^5+21 x^4+78 x^3+71 x^2+63 x+24$
- $y^2=34 x^6+79 x^5+73 x^4+52 x^3+8 x^2+78 x+44$
- $y^2=43 x^6+79 x^5+68 x^4+79 x^3+41 x^2+53 x+66$
- $y^2=30 x^6+78 x^5+61 x^4+36 x^3+78 x^2+26 x+77$
- $y^2=54 x^6+76 x^5+30 x^4+55 x^3+8 x^2+21 x+52$
- $y^2=66 x^6+18 x^5+40 x^4+15 x^3+13 x^2+4 x+20$
- $y^2=49 x^6+65 x^5+68 x^4+12 x^3+76 x^2+5 x+24$
- $y^2=50 x^6+68 x^5+30 x^4+26 x^3+6 x^2+73 x+47$
- $y^2=35 x^6+50 x^5+16 x^4+48 x^3+78 x^2+80 x+3$
- $y^2=38 x^6+30 x^5+57 x^4+45 x^3+67 x^2+39 x+54$
- $y^2=x^6+54 x^5+82 x^4+35 x^3+8 x^2+20 x+1$
- $y^2=66 x^6+29 x^5+74 x^4+68 x^3+56 x^2+57 x+80$
- $y^2=19 x^6+2 x^5+48 x^4+19 x^3+23 x^2+63 x+63$
- $y^2=8 x^6+24 x^5+72 x^3+56 x^2+8 x+67$
- $y^2=68 x^6+59 x^5+49 x^4+60 x^3+74 x^2+57 x+37$
- $y^2=72 x^6+13 x^5+14 x^4+41 x^3+40 x^2+18 x+4$
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 4.0.12255488.1. |
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 |
|---|---|---|
| 2.83.bc_nk | $2$ | (not in LMFDB) |