Invariants
| Base field: | $\F_{83}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 29 x + 363 x^{2} - 2407 x^{3} + 6889 x^{4}$ |
| Frobenius angles: | $\pm0.0299902267827$, $\pm0.296749112487$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.1766861.2 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4817$ | $46671913$ | $326922867971$ | $2252228443610669$ | $15515625846165633232$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $55$ | $6775$ | $571759$ | $47456979$ | $3938935200$ | $326938525435$ | $27136032642865$ | $2252292126331539$ | $186940255137421957$ | $15516041190410307750$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=49x^6+45x^5+27x^4+12x^3+35x^2+70x+4$
- $y^2=76x^6+42x^5+73x^4+37x^3+76x^2+59x+80$
- $y^2=36x^6+67x^5+34x^4+68x^3+61x^2+41x+63$
- $y^2=8x^6+28x^5+39x^4+54x^3+51x^2+2x+81$
- $y^2=71x^6+58x^5+22x^4+16x^3+3x^2+7x+68$
- $y^2=60x^6+75x^5+5x^4+42x^3+22x^2+27x+7$
- $y^2=47x^6+26x^5+33x^4+48x^3+28x^2+12x+14$
- $y^2=75x^6+13x^5+9x^4+52x^3+38x^2+24x+56$
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 4.0.1766861.2. |
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.bd_nz | $2$ | (not in LMFDB) |