Invariants
| Base field: | $\F_{73}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 13 x + 73 x^{2} )^{2}$ |
| $1 + 26 x + 315 x^{2} + 1898 x^{3} + 5329 x^{4}$ | |
| Frobenius angles: | $\pm0.775177233176$, $\pm0.775177233176$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $12$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3, 29$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $7569$ | $28164249$ | $150829703424$ | $807035542873641$ | $4297276623741592689$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $100$ | $5284$ | $387718$ | $28418500$ | $2072903140$ | $151334937358$ | $11047401572356$ | $806460000293764$ | $58871587676184214$ | $4297625823807468964$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=48 x^6+13 x^5+68 x^4+42 x^3+59 x^2+26 x+6$
- $y^2=41 x^6+57 x^5+45 x^4+25 x^3+58 x^2+55 x+12$
- $y^2=50 x^6+35 x^5+54 x^4+25 x^3+2 x^2+68 x+46$
- $y^2=43 x^6+29 x^5+22 x^4+9 x^3+43 x^2+49 x+61$
- $y^2=68 x^6+55 x^5+7 x^4+57 x^3+22 x^2+71 x+11$
- $y^2=50 x^6+27 x^5+57 x^4+26 x^3+17 x^2+48 x+18$
- $y^2=23 x^6+15 x^5+51 x^4+35 x^3+61 x^2+12 x+65$
- $y^2=6 x^6+53 x^5+61 x^4+13 x^3+72 x^2+41 x+69$
- $y^2=x^6+61 x^3+24$
- $y^2=4 x^6+43 x^5+5 x^4+27 x^3+10 x^2+26 x+32$
- $y^2=43 x^6+20 x^5+27 x^4+39 x^3+40 x^2+18 x+64$
- $y^2=20 x^6+9 x^5+51 x^4+68 x^3+9 x^2+12 x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{73}$.
Endomorphism algebra over $\F_{73}$| The isogeny class factors as 1.73.n 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-123}) \)$)$ |
Base change
This is a primitive isogeny class.