Invariants
| Base field: | $\F_{113}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 34 x + 507 x^{2} - 3842 x^{3} + 12769 x^{4}$ |
| Frobenius angles: | $\pm0.117489645955$, $\pm0.267759738399$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.945728.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $30$ |
| Isomorphism classes: | 30 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $9401$ | $161255353$ | $2083228057232$ | $26588110434772649$ | $339460522037225188201$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $80$ | $12628$ | $1443782$ | $163069860$ | $18424557120$ | $2081952536734$ | $235260545754272$ | $26584441955220420$ | $3004041940425679142$ | $339456739043156385588$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):
- $y^2=7 x^6+58 x^5+39 x^4+68 x^3+61 x^2+52 x+48$
- $y^2=30 x^6+94 x^5+72 x^4+38 x^3+13 x^2+105 x+67$
- $y^2=57 x^6+34 x^5+87 x^4+19 x^3+24 x^2+49 x+82$
- $y^2=111 x^6+59 x^5+2 x^4+23 x^3+10 x^2+104 x+24$
- $y^2=11 x^6+84 x^5+55 x^4+111 x^3+24 x^2+51 x+111$
- $y^2=55 x^6+10 x^5+19 x^4+82 x^3+35 x^2+92 x+80$
- $y^2=100 x^6+43 x^5+8 x^4+60 x^3+35 x^2+66 x+104$
- $y^2=24 x^6+3 x^5+83 x^4+82 x^3+15 x^2+17 x+8$
- $y^2=66 x^6+65 x^5+94 x^4+92 x^3+59 x^2+21 x+12$
- $y^2=94 x^6+63 x^5+28 x^4+90 x^3+60 x^2+13 x+63$
- $y^2=35 x^6+16 x^5+103 x^4+67 x^3+86 x^2+111 x+54$
- $y^2=44 x^6+97 x^5+88 x^4+28 x^3+16 x^2+41 x+55$
- $y^2=41 x^6+34 x^5+7 x^4+75 x^3+5 x^2+111 x+112$
- $y^2=3 x^6+91 x^5+26 x^4+18 x^3+78 x^2+96 x+3$
- $y^2=11 x^6+85 x^5+98 x^4+53 x^3+103 x^2+28 x+21$
- $y^2=80 x^6+45 x^5+34 x^4+104 x^3+51 x^2+47 x+60$
- $y^2=54 x^6+53 x^5+112 x^4+91 x^3+4 x^2+43 x+29$
- $y^2=53 x^6+77 x^5+5 x^4+55 x^3+73 x^2+56 x+94$
- $y^2=79 x^6+81 x^5+82 x^4+107 x^3+17 x^2+45 x+77$
- $y^2=58 x^6+99 x^5+78 x^4+41 x^3+24 x^2+17 x+88$
- $y^2=54 x^6+21 x^5+37 x^4+49 x^3+38 x^2+81 x+17$
- $y^2=105 x^6+100 x^5+15 x^4+46 x^3+45 x^2+38 x+85$
- $y^2=45 x^6+98 x^5+2 x^4+97 x^3+29 x^2+68 x+59$
- $y^2=79 x^6+41 x^5+67 x^4+70 x^3+108 x^2+87 x+21$
- $y^2=47 x^6+25 x^5+87 x^4+3 x^3+90 x^2+64 x+93$
- $y^2=111 x^6+74 x^5+39 x^4+67 x^3+5 x^2+76 x+110$
- $y^2=75 x^6+13 x^5+70 x^4+107 x^3+20 x^2+22 x+56$
- $y^2=55 x^6+61 x^5+72 x^4+59 x^3+67 x^2+63 x+42$
- $y^2=29 x^6+49 x^5+36 x^4+70 x^3+83 x^2+x+108$
- $y^2=14 x^6+105 x^5+88 x^4+55 x^3+47 x^2+103 x+87$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{113}$.
Endomorphism algebra over $\F_{113}$| The endomorphism algebra of this simple isogeny class is 4.0.945728.1. |
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 |
|---|---|---|
| 2.113.bi_tn | $2$ | (not in LMFDB) |