Invariants
| Base field: | $\F_{73}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 5 x + 73 x^{2} )^{2}$ |
| $1 + 10 x + 171 x^{2} + 730 x^{3} + 5329 x^{4}$ | |
| Frobenius angles: | $\pm0.594521390912$, $\pm0.594521390912$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $24$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $79$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $6241$ | $29713401$ | $150581250304$ | $806233944159081$ | $4298002000921457761$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $84$ | $5572$ | $387078$ | $28390276$ | $2073253044$ | $151333900558$ | $11047386901908$ | $806460173758468$ | $58871587147000854$ | $4297625821533792772$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=62 x^6+69 x^5+12 x^4+22 x^3+66 x^2+42 x+33$
- $y^2=16 x^6+23 x^5+23 x^4+52 x^3+50 x^2+40 x+13$
- $y^2=5 x^6+34 x^3+45$
- $y^2=4 x^6+60 x^5+10 x^4+16 x^3+29 x^2+61 x+70$
- $y^2=42 x^6+49 x^5+69 x^4+31 x^2+23 x+9$
- $y^2=47 x^6+23 x^5+28 x^4+34 x^3+2 x^2+25 x+18$
- $y^2=12 x^6+44 x^5+47 x^4+58 x^3+14 x^2+62 x+69$
- $y^2=42 x^6+57 x^5+50 x^4+24 x^3+17 x^2+16 x+50$
- $y^2=31 x^6+26 x^5+50 x^4+6 x^3+12 x^2+2 x+51$
- $y^2=64 x^6+52 x^5+71 x^4+20 x^3+40 x^2+70 x+62$
- $y^2=54 x^6+29 x^5+67 x^4+9 x^3+37 x^2+59 x+69$
- $y^2=67 x^6+x^5+32 x^4+71 x^3+34 x^2+36 x+46$
- $y^2=69 x^6+34 x^5+55 x^4+14 x^3+x^2+33 x+36$
- $y^2=11 x^6+37 x^5+52 x^4+18 x^3+65 x^2+2 x+57$
- $y^2=9 x^6+71 x^5+8 x^4+61 x^3+7 x^2+48 x+50$
- $y^2=8 x^6+18 x^5+70 x^4+47 x^3+54 x^2+49 x+12$
- $y^2=18 x^6+32 x^5+53 x^4+51 x^2+65 x+6$
- $y^2=45 x^6+64 x^5+12 x^4+12 x^3+6 x^2+16 x+33$
- $y^2=50 x^6+63 x^5+37 x^4+21 x^3+2 x^2+47 x+52$
- $y^2=52 x^6+41 x^5+19 x^4+45 x^3+34 x^2+40 x+3$
- $y^2=66 x^6+32 x^5+48 x^4+16 x^3+70 x^2+47 x+36$
- $y^2=60 x^6+48 x^5+12 x^4+26 x^3+28 x^2+53 x+43$
- $y^2=10 x^6+34 x^5+40 x^4+11 x^3+25 x^2+9 x+71$
- $y^2=10 x^6+23 x^5+55 x^4+61 x^3+31 x^2+27 x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{73}$.
Endomorphism algebra over $\F_{73}$| The isogeny class factors as 1.73.f 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-267}) \)$)$ |
Base change
This is a primitive isogeny class.