Invariants
| Base field: | $\F_{73}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 4 x + 148 x^{2} - 292 x^{3} + 5329 x^{4}$ |
| Frobenius angles: | $\pm0.435970192556$, $\pm0.489086021060$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.5232896.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $30$ |
| Isomorphism classes: | 30 |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $5182$ | $29920868$ | $151660314046$ | $805950535067024$ | $4297434192014045182$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $70$ | $5610$ | $389854$ | $28380294$ | $2072979150$ | $151335264810$ | $11047406659030$ | $806460039404094$ | $58871586089479942$ | $4297625831839511050$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):
- $y^2=45 x^5+41 x^4+14 x^3+48 x^2+52 x+1$
- $y^2=26 x^6+34 x^5+71 x^4+43 x^3+37 x^2+71 x+70$
- $y^2=32 x^6+16 x^4+x^3+53 x^2+15 x+68$
- $y^2=20 x^6+27 x^5+70 x^4+34 x^3+60 x^2+60 x+72$
- $y^2=41 x^6+8 x^5+59 x^4+50 x^3+45 x^2+29 x+72$
- $y^2=21 x^6+51 x^5+44 x^4+37 x^3+67 x^2+27 x+35$
- $y^2=23 x^6+68 x^5+17 x^4+2 x^3+4 x^2+17 x+71$
- $y^2=56 x^6+25 x^5+18 x^4+24 x^3+x^2+71 x+24$
- $y^2=51 x^6+65 x^5+42 x^4+17 x^3+15 x^2+19 x+45$
- $y^2=20 x^6+67 x^5+8 x^4+21 x^3+66 x^2+44 x+49$
- $y^2=54 x^6+59 x^5+61 x^4+11 x^3+8 x^2+60 x+67$
- $y^2=29 x^6+45 x^5+21 x^4+32 x^3+69 x^2+15 x+42$
- $y^2=33 x^6+14 x^5+57 x^4+5 x^3+7 x^2+18 x+42$
- $y^2=54 x^6+72 x^5+3 x^4+43 x^3+23 x^2+x+56$
- $y^2=53 x^6+41 x^5+41 x^4+14 x^3+72 x^2+2 x+7$
- $y^2=17 x^6+63 x^5+36 x^4+10 x^3+x^2+49 x+43$
- $y^2=14 x^6+13 x^5+57 x^4+64 x^3+35 x^2+22 x+62$
- $y^2=31 x^6+18 x^5+57 x^4+50 x^3+20 x^2+9 x+57$
- $y^2=63 x^6+8 x^5+67 x^4+49 x^3+71 x^2+5 x+6$
- $y^2=10 x^6+72 x^5+63 x^4+28 x^3+6 x^2+51 x+37$
- $y^2=69 x^6+12 x^5+11 x^4+66 x^3+52 x^2+3 x+13$
- $y^2=57 x^6+11 x^5+4 x^4+9 x^3+6 x^2+60 x+59$
- $y^2=65 x^6+41 x^5+34 x^4+71 x^3+40 x^2+7 x+36$
- $y^2=5 x^6+17 x^5+22 x^4+36 x^3+58 x^2+49 x+70$
- $y^2=63 x^6+72 x^5+52 x^4+30 x^3+25 x^2+26 x+56$
- $y^2=54 x^6+39 x^5+57 x^4+54 x^3+44 x^2+x+55$
- $y^2=29 x^6+9 x^5+25 x^4+12 x^3+40 x^2+19 x+30$
- $y^2=10 x^6+63 x^5+58 x^4+46 x^3+23 x^2+22 x+66$
- $y^2=16 x^6+x^5+15 x^4+64 x^3+11 x^2+55 x+35$
- $y^2=35 x^6+47 x^5+64 x^4+12 x^3+66 x^2+33 x+51$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{73}$.
Endomorphism algebra over $\F_{73}$| The endomorphism algebra of this simple isogeny class is 4.0.5232896.2. |
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 |
|---|---|---|
| 2.73.e_fs | $2$ | (not in LMFDB) |