Invariants
Base field: | $\F_{83}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 18 x + 83 x^{2} )( 1 - 16 x + 83 x^{2} )$ |
$1 - 34 x + 454 x^{2} - 2822 x^{3} + 6889 x^{4}$ | |
Frobenius angles: | $\pm0.0496118990883$, $\pm0.158801688027$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $6$ |
Isomorphism classes: | 16 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $4488$ | $45777600$ | $326105714088$ | $2252030863104000$ | $15516083908976100648$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $50$ | $6642$ | $570326$ | $47452814$ | $3939051490$ | $326940825474$ | $27136055965222$ | $2252292264606046$ | $186940255313618258$ | $15516041184959062482$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=3x^6+22x^5+27x^4+77x^3+9x^2+67x+37$
- $y^2=20x^6+12x^5+65x^4+50x^3+65x^2+12x+20$
- $y^2=82x^6+14x^5+37x^4+66x^3+21x^2+19x+34$
- $y^2=15x^6+32x^5+16x^4+73x^3+16x^2+32x+15$
- $y^2=59x^6+81x^5+79x^4+58x^3+39x^2+7x+11$
- $y^2=59x^6+62x^5+53x^4+9x^3+53x^2+62x+59$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{83}$.
Endomorphism algebra over $\F_{83}$The isogeny class factors as 1.83.as $\times$ 1.83.aq 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_aes | $2$ | (not in LMFDB) |
2.83.c_aes | $2$ | (not in LMFDB) |
2.83.bi_rm | $2$ | (not in LMFDB) |