Invariants
| Base field: | $\F_{73}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 17 x + 214 x^{2} + 1241 x^{3} + 5329 x^{4}$ |
| Frobenius angles: | $\pm0.622969282568$, $\pm0.712084421304$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-862 +34 \sqrt{17}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $24$ |
| Isomorphism classes: | 24 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $6802$ | $29153372$ | $150450118624$ | $806721381543104$ | $4297766836496429122$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $91$ | $5469$ | $386740$ | $28407441$ | $2073139611$ | $151333187682$ | $11047402343299$ | $806460115714209$ | $58871586411396676$ | $4297625830441770109$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=54 x^6+15 x^5+26 x^4+70 x^3+36 x^2+66 x+49$
- $y^2=32 x^6+29 x^5+31 x^4+3 x^3+24 x^2+56 x+49$
- $y^2=63 x^6+66 x^5+35 x^4+70 x^3+22 x^2+60 x+4$
- $y^2=56 x^6+19 x^5+45 x^4+57 x^3+17 x^2+39 x+32$
- $y^2=49 x^6+x^5+11 x^4+22 x^3+47 x^2+6 x+63$
- $y^2=42 x^6+16 x^5+16 x^4+64 x^3+51 x^2+54 x+69$
- $y^2=53 x^6+4 x^5+65 x^4+60 x^3+30 x^2+63 x+33$
- $y^2=21 x^6+42 x^5+29 x^4+51 x^3+22 x^2+59 x+8$
- $y^2=55 x^6+34 x^5+9 x^4+9 x^3+35 x^2+21 x+55$
- $y^2=7 x^6+34 x^5+36 x^4+29 x^3+54 x^2+61 x+60$
- $y^2=17 x^6+58 x^5+63 x^4+56 x^3+5 x^2+35 x+3$
- $y^2=57 x^6+39 x^5+41 x^4+46 x^3+32 x^2+15 x+42$
- $y^2=69 x^6+19 x^5+4 x^4+59 x^3+44 x^2+3 x+17$
- $y^2=10 x^6+3 x^5+63 x^4+56 x^3+29 x^2+9 x+61$
- $y^2=38 x^6+4 x^5+23 x^4+21 x^3+57 x^2+62 x+5$
- $y^2=26 x^6+56 x^5+67 x^4+36 x^3+36 x^2+24 x+56$
- $y^2=67 x^6+29 x^5+56 x^4+61 x^3+21 x^2+9 x+8$
- $y^2=52 x^6+4 x^5+20 x^4+21 x^3+69 x^2+4 x+57$
- $y^2=68 x^6+10 x^5+9 x^4+67 x^3+37 x+1$
- $y^2=34 x^6+9 x^5+44 x^4+8 x^3+6 x^2+56 x+71$
- $y^2=2 x^6+41 x^5+12 x^4+61 x^3+56 x^2+10 x+24$
- $y^2=45 x^6+50 x^5+59 x^4+51 x^3+17 x^2+36 x+35$
- $y^2=15 x^6+39 x^5+62 x^4+3 x^3+68 x^2+53 x+50$
- $y^2=8 x^6+65 x^5+34 x^4+24 x^3+51 x^2+40 x+34$
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 \(\Q(\sqrt{-862 +34 \sqrt{17}})\). |
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.ar_ig | $2$ | (not in LMFDB) |