Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 23 x + 239 x^{2} - 1403 x^{3} + 3721 x^{4}$ |
| Frobenius angles: | $\pm0.0529194219962$, $\pm0.338378256587$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-386 -46 \sqrt{61}})\) |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2535$ | $13656045$ | $51546134835$ | $191672000590005$ | $713281995600954000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $39$ | $3671$ | $227097$ | $13843291$ | $844524174$ | $51519676223$ | $3142740010449$ | $191707323161011$ | $11694146290811127$ | $713342912420718326$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=42 x^6+9 x^5+54 x^4+9 x^3+54 x^2+24 x+35$
- $y^2=51 x^6+16 x^5+39 x^4+53 x^2+40 x+35$
- $y^2=47 x^6+12 x^5+16 x^4+26 x^3+32 x^2+14 x+28$
- $y^2=24 x^6+45 x^5+49 x^4+60 x^3+29 x^2+2 x+55$
- $y^2=53 x^6+57 x^5+31 x^4+2 x^3+14 x^2+13 x+59$
- $y^2=21 x^6+14 x^5+27 x^4+51 x^3+8 x^2+38 x+11$
- $y^2=28 x^6+13 x^5+25 x^4+51 x^3+8 x^2+16 x+8$
- $y^2=24 x^6+17 x^5+9 x^4+58 x^3+49 x^2+50 x+37$
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 \(\Q(\sqrt{-386 -46 \sqrt{61}})\). |
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.x_jf | $2$ | (not in LMFDB) |