Invariants
| Base field: | $\F_{79}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 30 x + 380 x^{2} + 2370 x^{3} + 6241 x^{4}$ |
| Frobenius angles: | $\pm0.768210038880$, $\pm0.890348806862$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-88 -30 \sqrt{3}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $9022$ | $38090884$ | $243043486582$ | $1517489167321296$ | $9467924844424720102$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $110$ | $6102$ | $492950$ | $38959846$ | $3076942250$ | $243088256742$ | $19203905747570$ | $1517108812199038$ | $119851595958026990$ | $9468276085168490502$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=72 x^6+x^5+75 x^4+53 x^3+59 x^2+11 x+50$
- $y^2=47 x^6+47 x^5+49 x^4+27 x^3+10 x^2+70 x+20$
- $y^2=38 x^6+70 x^5+47 x^4+42 x^3+34 x^2+20 x+78$
- $y^2=20 x^6+24 x^5+75 x^4+20 x^3+x^2+x+59$
- $y^2=21 x^6+51 x^5+40 x^4+60 x^3+64 x^2+15 x+18$
- $y^2=6 x^6+38 x^5+52 x^4+12 x^3+69 x^2+51 x+39$
- $y^2=27 x^6+64 x^5+70 x^4+59 x^3+2 x^2+27 x+4$
- $y^2=17 x^6+76 x^5+28 x^4+57 x^3+51 x^2+x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{79}$.
Endomorphism algebra over $\F_{79}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-88 -30 \sqrt{3}})\). |
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 |
|---|---|---|
| 2.79.abe_oq | $2$ | (not in LMFDB) |