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: | $\pm0.224822766824$, $\pm0.224822766824$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $12$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3721$ | $28164249$ | $151841150224$ | $807035542873641$ | $4297975058146184041$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $48$ | $5284$ | $390318$ | $28418500$ | $2073240048$ | $151334937358$ | $11047395465840$ | $806460000293764$ | $58871585740351614$ | $4297625823807468964$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=21 x^6+65 x^5+48 x^4+64 x^3+3 x^2+57 x+30$
- $y^2=59 x^6+66 x^5+6 x^4+52 x^3+71 x^2+56 x+60$
- $y^2=10 x^6+7 x^5+40 x^4+5 x^3+15 x^2+72 x+53$
- $y^2=69 x^6+72 x^5+37 x^4+45 x^3+69 x^2+26 x+13$
- $y^2=48 x^6+56 x^5+35 x^4+66 x^3+37 x^2+63 x+55$
- $y^2=10 x^6+20 x^5+26 x^4+49 x^3+18 x^2+68 x+62$
- $y^2=63 x^6+3 x^5+54 x^4+7 x^3+56 x^2+17 x+13$
- $y^2=45 x^6+69 x^5+56 x^4+61 x^3+29 x^2+52 x+43$
- $y^2=5 x^6+13 x^3+47$
- $y^2=20 x^6+69 x^5+25 x^4+62 x^3+50 x^2+57 x+14$
- $y^2=69 x^6+27 x^5+62 x^4+49 x^3+54 x^2+17 x+28$
- $y^2=27 x^6+45 x^5+36 x^4+48 x^3+45 x^2+60 x+10$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{73}$.
Endomorphism algebra over $\F_{73}$| The isogeny class factors as 1.73.an 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-123}) \)$)$ |
Base change
This is a primitive isogeny class.