Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 25 x + 273 x^{2} - 1525 x^{3} + 3721 x^{4}$ |
| Frobenius angles: | $\pm0.104181476439$, $\pm0.273280674769$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-330 +50 \sqrt{21}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2445$ | $13557525$ | $51582822705$ | $191779339197525$ | $713366139840738000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $37$ | $3643$ | $227257$ | $13851043$ | $844623802$ | $51520355323$ | $3142741760797$ | $191707313898883$ | $11694146277548617$ | $713342914465733398$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=28 x^6+36 x^5+3 x^4+46 x^3+52 x^2+39 x+32$
- $y^2=48 x^6+5 x^5+41 x^4+32 x^3+13 x^2+38 x+29$
- $y^2=48 x^6+41 x^5+29 x^4+57 x^3+32 x^2+35 x+35$
- $y^2=60 x^6+24 x^5+57 x^4+6 x^3+27 x^2+14 x+51$
- $y^2=26 x^6+17 x^5+31 x^4+12 x^3+57 x^2+27 x+58$
- $y^2=37 x^6+18 x^5+20 x^4+10 x^3+57 x^2+51 x+7$
- $y^2=33 x^6+20 x^5+39 x^4+25 x^3+52 x^2+4 x+8$
- $y^2=10 x^6+29 x^5+57 x^4+39 x^3+50 x^2+31 x+36$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{61}$.
Endomorphism algebra over $\F_{61}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-330 +50 \sqrt{21}})\). |
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 |
|---|---|---|
| 2.61.z_kn | $2$ | (not in LMFDB) |