Invariants
| Base field: | $\F_{109}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 34 x + 497 x^{2} - 3706 x^{3} + 11881 x^{4}$ |
| Frobenius angles: | $\pm0.0837355935881$, $\pm0.269407611197$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.142400.3 |
| Galois group: | $D_{4}$ |
| Jacobians: | $18$ |
| Isomorphism classes: | 36 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $8639$ | $139252041$ | $1677451942556$ | $19927218974042169$ | $236737167351843864599$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $76$ | $11720$ | $1295302$ | $141169444$ | $15386291536$ | $1677099203966$ | $182803893990304$ | $19925626311131524$ | $2171893280858996638$ | $236736367503674969000$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):
- $y^2=52 x^6+40 x^5+12 x^4+30 x^3+67 x^2+30 x+24$
- $y^2=61 x^6+40 x^5+40 x^4+90 x^3+87 x^2+66 x+38$
- $y^2=37 x^6+11 x^5+66 x^4+86 x^3+30 x^2+67 x+72$
- $y^2=76 x^6+82 x^5+27 x^4+65 x^3+78 x^2+28 x+18$
- $y^2=20 x^6+12 x^5+17 x^4+107 x^3+56 x^2+108 x+20$
- $y^2=63 x^6+x^5+51 x^4+18 x^3+101 x^2+72 x+108$
- $y^2=82 x^6+12 x^5+17 x^4+9 x^3+50 x^2+32 x+67$
- $y^2=67 x^6+39 x^5+72 x^4+63 x^3+38 x^2+86 x+55$
- $y^2=11 x^6+6 x^5+12 x^4+55 x^3+54 x^2+63 x+39$
- $y^2=102 x^6+38 x^5+88 x^4+85 x^3+72 x^2+89 x+83$
- $y^2=2 x^6+86 x^5+98 x^4+74 x^3+45 x^2+41 x+11$
- $y^2=9 x^6+2 x^5+84 x^4+53 x^3+38 x^2+74 x+28$
- $y^2=7 x^6+16 x^5+94 x^4+54 x^3+63 x^2+40 x+76$
- $y^2=82 x^6+31 x^5+62 x^4+20 x^3+62 x^2+43 x+50$
- $y^2=3 x^6+15 x^5+68 x^4+26 x^3+79 x^2+107 x+23$
- $y^2=106 x^6+108 x^5+48 x^4+84 x^3+16 x^2+43 x+88$
- $y^2=91 x^6+30 x^5+104 x^4+41 x^3+88 x^2+66 x+54$
- $y^2=80 x^6+54 x^5+34 x^4+62 x^3+66 x^2+21 x+21$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{109}$.
Endomorphism algebra over $\F_{109}$| The endomorphism algebra of this simple isogeny class is 4.0.142400.3. |
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 |
|---|---|---|
| 2.109.bi_td | $2$ | (not in LMFDB) |