Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 22 x + 235 x^{2} - 1342 x^{3} + 3721 x^{4}$ |
| Frobenius angles: | $\pm0.153970557912$, $\pm0.324763714073$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.598592.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $18$ |
| Isomorphism classes: | 18 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2593$ | $13797353$ | $51710477248$ | $191804854746857$ | $713361103306533433$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $40$ | $3708$ | $227818$ | $13852884$ | $844617840$ | $51520367334$ | $3142743959584$ | $191707342067940$ | $11694146378777074$ | $713342912655993868$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):
- $y^2=21 x^6+59 x^5+41 x^4+55 x^3+39 x^2+44 x+25$
- $y^2=50 x^6+40 x^5+16 x^4+33 x^3+19 x^2+59 x+10$
- $y^2=37 x^6+40 x^4+24 x^3+16 x^2+53 x+43$
- $y^2=31 x^6+32 x^5+21 x^4+17 x^3+10 x^2+50 x+33$
- $y^2=6 x^6+38 x^5+31 x^4+9 x^3+57 x^2+27 x+23$
- $y^2=20 x^6+19 x^5+2 x^4+16 x^3+3 x^2+26 x+59$
- $y^2=14 x^6+57 x^5+44 x^4+54 x^3+49 x+1$
- $y^2=58 x^6+6 x^5+12 x^4+38 x^3+55 x^2+19 x+49$
- $y^2=47 x^6+18 x^5+44 x^4+5 x^3+36 x^2+9 x+51$
- $y^2=32 x^6+44 x^5+37 x^4+17 x^3+2 x^2+23 x+34$
- $y^2=23 x^6+54 x^5+12 x^4+42 x^3+5 x^2+43 x+37$
- $y^2=51 x^6+30 x^5+27 x^4+21 x^3+16 x^2+35 x+26$
- $y^2=46 x^6+43 x^5+4 x^4+56 x^3+54 x^2+53 x+10$
- $y^2=x^6+49 x^5+10 x^4+32 x^3+34 x^2+9 x+51$
- $y^2=7 x^6+56 x^5+24 x^4+52 x^3+51 x^2+16 x+28$
- $y^2=38 x^6+24 x^5+58 x^4+57 x^3+53 x^2+34 x+8$
- $y^2=54 x^6+20 x^5+33 x^4+5 x^3+37 x^2+34 x+40$
- $y^2=26 x^6+60 x^5+31 x^4+9 x^3+46 x^2+29 x+48$
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 4.0.598592.1. |
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.w_jb | $2$ | (not in LMFDB) |