Invariants
| Base field: | $\F_{67}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 119 x^{2} + 4489 x^{4}$ |
| Frobenius angles: | $\pm0.0760265737172$, $\pm0.923973426283$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-15}, \sqrt{253})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $24$ |
| Isomorphism classes: | 32 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4371$ | $19105641$ | $90458299584$ | $405858858191721$ | $1822837806521701611$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $68$ | $4252$ | $300764$ | $20140756$ | $1350125108$ | $90458216998$ | $6060711605324$ | $406067704434148$ | $27206534396294948$ | $1822837808491641772$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=22 x^6+37 x^5+7 x^4+25 x^3+21 x^2+65 x+58$
- $y^2=60 x^6+10 x^5+30 x^4+34 x^3+25 x^2+41 x+30$
- $y^2=53 x^6+20 x^5+60 x^4+x^3+50 x^2+15 x+60$
- $y^2=2 x^6+2 x^5+29 x^4+5 x^3+30 x^2+7 x+65$
- $y^2=28 x^6+33 x^5+8 x^4+5 x^3+33 x^2+x+15$
- $y^2=56 x^6+66 x^5+16 x^4+10 x^3+66 x^2+2 x+30$
- $y^2=38 x^6+10 x^5+63 x^4+40 x^3+15 x^2+15 x+4$
- $y^2=17 x^6+22 x^5+14 x^4+63 x^3+12 x^2+23 x+57$
- $y^2=20 x^6+28 x^5+34 x^4+21 x^3+54 x^2+12 x+5$
- $y^2=40 x^6+56 x^5+x^4+42 x^3+41 x^2+24 x+10$
- $y^2=5 x^6+53 x^5+34 x^3+32 x^2+53 x+2$
- $y^2=10 x^6+39 x^5+x^3+64 x^2+39 x+4$
- $y^2=4 x^6+9 x^5+14 x^4+61 x^3+52 x^2+64 x+11$
- $y^2=17 x^6+21 x^5+54 x^4+22 x^3+52 x^2+x+51$
- $y^2=27 x^6+66 x^4+12 x^3+22 x^2+1$
- $y^2=61 x^6+13 x^4+27 x^3+33 x^2+49$
- $y^2=33 x^6+60 x^5+35 x^4+43 x^3+54 x^2+2 x+28$
- $y^2=66 x^6+53 x^5+3 x^4+19 x^3+41 x^2+4 x+56$
- $y^2=13 x^6+60 x^5+33 x^4+41 x^3+2 x^2+36 x+13$
- $y^2=26 x^6+53 x^5+66 x^4+15 x^3+4 x^2+5 x+26$
- $y^2=28 x^6+57 x^5+30 x^4+4 x^3+10 x^2+18 x+65$
- $y^2=56 x^6+47 x^5+60 x^4+8 x^3+20 x^2+36 x+63$
- $y^2=21 x^6+5 x^5+50 x^4+61 x^3+51 x^2+2 x+48$
- $y^2=42 x^6+10 x^5+33 x^4+55 x^3+35 x^2+4 x+29$
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.aep 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_ep | $4$ | (not in LMFDB) |