Invariants
Base field: | $\F_{83}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 28 x + 347 x^{2} - 2324 x^{3} + 6889 x^{4}$ |
Frobenius angles: | $\pm0.0623004423497$, $\pm0.312415000213$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.10371600.2 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 16 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $4885$ | $46842265$ | $327067730260$ | $2252291567383225$ | $15515669398619867125$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $56$ | $6800$ | $572012$ | $47458308$ | $3938946256$ | $326938875950$ | $27136040238832$ | $2252292231460228$ | $186940256149294676$ | $15516041197461398000$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=82x^6+36x^5+64x^4+62x^3+54x^2+70x+22$
- $y^2=58x^6+72x^5+55x^4+43x^3+69x^2+40x+24$
- $y^2=46x^6+80x^5+35x^4+18x^3+32x^2+x+73$
- $y^2=53x^6+24x^5+32x^4+56x^3+17x^2+46x+53$
- $y^2=74x^6+61x^5+36x^4+76x^3+24x^2+73x+69$
- $y^2=14x^6+73x^5+27x^4+46x^3+19x^2+44x+72$
- $y^2=22x^6+62x^5+62x^4+72x^3+75x^2+12x+2$
- $y^2=52x^6+10x^5+68x^4+62x^3+31x^2+20x+53$
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 4.0.10371600.2. |
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.bc_nj | $2$ | (not in LMFDB) |