Invariants
Base field: | $\F_{73}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 17 x + 73 x^{2} )( 1 - 11 x + 73 x^{2} )$ |
$1 - 28 x + 333 x^{2} - 2044 x^{3} + 5329 x^{4}$ | |
Frobenius angles: | $\pm0.0323195869136$, $\pm0.277387524567$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $13$ |
Isomorphism classes: | 42 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3591$ | $27776385$ | $151290151488$ | $806466922832025$ | $4297526514615651471$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $46$ | $5212$ | $388906$ | $28398484$ | $2073023686$ | $151333204174$ | $11047386944710$ | $806460008962276$ | $58871586406545178$ | $4297625830660419532$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 13 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=20x^6+18x^5+7x^4+36x^3+15x^2+9x+7$
- $y^2=47x^6+5x^5+10x^4+39x^3+12x+19$
- $y^2=42x^6+56x^5+60x^4+13x^3+69x^2+70x+56$
- $y^2=40x^6+63x^5+2x^4+10x^3+2x^2+63x+40$
- $y^2=48x^6+38x^5+58x^4+47x^3+58x^2+38x+48$
- $y^2=20x^6+10x^5+54x^4+26x^3+8x^2+58x+14$
- $y^2=54x^6+64x^5+15x^4+16x^3+19x^2+42x+45$
- $y^2=x^6+49x^5+57x^4+67x^3+28x^2+67x+41$
- $y^2=28x^6+24x^5+12x^4+12x^3+25x^2+34$
- $y^2=58x^6+32x^5+72x^4+31x^3+72x^2+32x+58$
- $y^2=32x^6+27x^5+42x^4+39x^3+42x^2+27x+32$
- $y^2=25x^6+25x^5+44x^4+60x^3+61x^2+60x+71$
- $y^2=52x^6+70x^5+19x^4+38x^3+19x^2+70x+52$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{73}$.
Endomorphism algebra over $\F_{73}$The isogeny class factors as 1.73.ar $\times$ 1.73.al 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.