Invariants
| Base field: | $\F_{67}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 119 x^{2} + 4489 x^{4}$ |
| Frobenius angles: | $\pm0.423973426283$, $\pm0.576026573717$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{15}, \sqrt{-253})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $24$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4609$ | $21242881$ | $90458464756$ | $405858858191721$ | $1822837802581821289$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $68$ | $4728$ | $300764$ | $20140756$ | $1350125108$ | $90458547342$ | $6060711605324$ | $406067704434148$ | $27206534396294948$ | $1822837800611881128$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=2 x^6+13 x^5+8 x^4+54 x^3+66 x^2+34 x+56$
- $y^2=4 x^6+26 x^5+16 x^4+41 x^3+65 x^2+x+45$
- $y^2=24 x^6+51 x^5+54 x^4+27 x^3+24 x^2+35 x+2$
- $y^2=42 x^6+63 x^5+19 x^4+59 x^3+17 x^2+48 x+43$
- $y^2=17 x^6+59 x^5+38 x^4+51 x^3+34 x^2+29 x+19$
- $y^2=51 x^6+13 x^5+23 x^4+52 x^3+25 x^2+60 x+32$
- $y^2=64 x^6+42 x^5+19 x^4+62 x^3+65 x^2+32 x+31$
- $y^2=6 x^6+49 x^5+59 x^4+63 x^3+4 x^2+36$
- $y^2=12 x^6+31 x^5+51 x^4+59 x^3+8 x^2+5$
- $y^2=61 x^6+8 x^5+47 x^4+9 x^3+17 x^2+27 x+5$
- $y^2=45 x^6+16 x^5+12 x^4+44 x^3+22 x^2+5 x+11$
- $y^2=34 x^6+30 x^5+8 x^4+47 x^3+16 x^2+18 x+41$
- $y^2=21 x^6+28 x^5+21 x^4+39 x^3+13 x^2+56 x+35$
- $y^2=5 x^6+42 x^5+8 x^4+56 x^3+4 x^2+27 x+4$
- $y^2=10 x^6+17 x^5+16 x^4+45 x^3+8 x^2+54 x+8$
- $y^2=8 x^6+13 x^5+47 x^4+60 x^3+45 x^2+8 x+45$
- $y^2=48 x^6+40 x^5+59 x^4+26 x^3+36 x^2+57 x+42$
- $y^2=29 x^6+13 x^5+51 x^4+52 x^3+5 x^2+47 x+17$
- $y^2=36 x^6+24 x^5+63 x^4+10 x^3+3 x^2+23 x+36$
- $y^2=5 x^6+48 x^5+59 x^4+20 x^3+6 x^2+46 x+5$
- $y^2=24 x^6+19 x^5+42 x^4+43 x^3+29 x^2+3 x+46$
- $y^2=48 x^6+38 x^5+17 x^4+19 x^3+58 x^2+6 x+25$
- $y^2=51 x^6+63 x^5+47 x^4+14 x^3+6 x^2+50 x+4$
- $y^2=35 x^6+59 x^5+27 x^4+28 x^3+12 x^2+33 x+8$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{67^{2}}$.
Endomorphism algebra over $\F_{67}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{15}, \sqrt{-253})\). |
| The base change of $A$ to $\F_{67^{2}}$ is 1.4489.ep 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3795}) \)$)$ |
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 |
|---|---|---|
| 2.67.a_aep | $4$ | (not in LMFDB) |