Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 89 x^{2} + 3721 x^{4}$ |
| Frobenius angles: | $\pm0.119874499213$, $\pm0.880125500787$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-33}, \sqrt{211})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $18$ |
| Isomorphism classes: | 24 |
| Cyclic group of points: | yes |
This isogeny class is simple but not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3633$ | $13198689$ | $51520662900$ | $191694076601769$ | $713342913033372153$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $62$ | $3544$ | $226982$ | $13844884$ | $844596302$ | $51520951438$ | $3142742836022$ | $191707367921764$ | $11694146092834142$ | $713342914403861704$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):
- $y^2=43 x^6+23 x^5+6 x^4+13 x^3+34 x^2+54 x+39$
- $y^2=22 x^6+14 x^5+17 x^4+18 x^3+49 x^2+58 x+24$
- $y^2=44 x^6+28 x^5+34 x^4+36 x^3+37 x^2+55 x+48$
- $y^2=27 x^6+9 x^5+45 x^4+37 x^3+40 x^2+41 x+28$
- $y^2=58 x^6+60 x^5+12 x^4+7 x^3+26 x^2+58 x+11$
- $y^2=21 x^6+11 x^5+22 x^4+7 x^3+43 x^2+24 x+22$
- $y^2=51 x^6+55 x^5+6 x^4+4 x^3+58 x^2+29 x+47$
- $y^2=46 x^6+2 x^5+43 x^4+30 x^3+7 x^2+34 x+10$
- $y^2=31 x^6+4 x^5+25 x^4+60 x^3+14 x^2+7 x+20$
- $y^2=24 x^6+59 x^5+41 x^4+34 x^3+59 x^2+50 x+1$
- $y^2=42 x^6+14 x^5+8 x^4+39 x^3+43 x^2+8 x+9$
- $y^2=23 x^6+28 x^5+16 x^4+17 x^3+25 x^2+16 x+18$
- $y^2=41 x^6+25 x^5+25 x^4+22 x^3+41 x^2+48 x+38$
- $y^2=21 x^6+50 x^5+50 x^4+44 x^3+21 x^2+35 x+15$
- $y^2=46 x^6+7 x^5+51 x^4+44 x^3+39 x^2+37 x+39$
- $y^2=31 x^6+14 x^5+41 x^4+27 x^3+17 x^2+13 x+17$
- $y^2=15 x^6+30 x^5+59 x^4+10 x^3+55 x^2+34 x+23$
- $y^2=30 x^6+60 x^5+57 x^4+20 x^3+49 x^2+7 x+46$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{61^{2}}$.
Endomorphism algebra over $\F_{61}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-33}, \sqrt{211})\). |
| The base change of $A$ to $\F_{61^{2}}$ is 1.3721.adl 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6963}) \)$)$ |
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.a_dl | $4$ | (not in LMFDB) |