Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 664*x^14 - 3888*x^13 + 167028*x^12 - 646096*x^11 + 19823736*x^10 - 38554608*x^9 + 1128716344*x^8 - 144917456*x^7 + 29145288616*x^6 + 33165335168*x^5 + 345115392384*x^4 + 603518307936*x^3 + 2078152822832*x^2 + 2472153157008*x + 3733330807202)
gp: K = bnfinit(y^16 - 8*y^15 + 664*y^14 - 3888*y^13 + 167028*y^12 - 646096*y^11 + 19823736*y^10 - 38554608*y^9 + 1128716344*y^8 - 144917456*y^7 + 29145288616*y^6 + 33165335168*y^5 + 345115392384*y^4 + 603518307936*y^3 + 2078152822832*y^2 + 2472153157008*y + 3733330807202, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 664*x^14 - 3888*x^13 + 167028*x^12 - 646096*x^11 + 19823736*x^10 - 38554608*x^9 + 1128716344*x^8 - 144917456*x^7 + 29145288616*x^6 + 33165335168*x^5 + 345115392384*x^4 + 603518307936*x^3 + 2078152822832*x^2 + 2472153157008*x + 3733330807202);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 664*x^14 - 3888*x^13 + 167028*x^12 - 646096*x^11 + 19823736*x^10 - 38554608*x^9 + 1128716344*x^8 - 144917456*x^7 + 29145288616*x^6 + 33165335168*x^5 + 345115392384*x^4 + 603518307936*x^3 + 2078152822832*x^2 + 2472153157008*x + 3733330807202)
\( x^{16} - 8 x^{15} + 664 x^{14} - 3888 x^{13} + 167028 x^{12} - 646096 x^{11} + 19823736 x^{10} + \cdots + 3733330807202 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(6896992068684009105950840251291271168\)
\(\medspace = 2^{62}\cdot 113^{3}\cdot 1009^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(200.64\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(113\), \(1009\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{113}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{62\!\cdots\!73}a^{15}-\frac{92\!\cdots\!33}{62\!\cdots\!73}a^{14}-\frac{52\!\cdots\!79}{62\!\cdots\!73}a^{13}-\frac{25\!\cdots\!86}{62\!\cdots\!73}a^{12}-\frac{17\!\cdots\!04}{62\!\cdots\!73}a^{11}-\frac{86\!\cdots\!18}{20\!\cdots\!83}a^{10}-\frac{22\!\cdots\!80}{62\!\cdots\!73}a^{9}+\frac{30\!\cdots\!39}{62\!\cdots\!73}a^{8}-\frac{47\!\cdots\!50}{62\!\cdots\!73}a^{7}+\frac{29\!\cdots\!56}{13\!\cdots\!59}a^{6}-\frac{27\!\cdots\!61}{62\!\cdots\!73}a^{5}-\frac{25\!\cdots\!62}{62\!\cdots\!73}a^{4}+\frac{14\!\cdots\!33}{62\!\cdots\!73}a^{3}+\frac{23\!\cdots\!85}{62\!\cdots\!73}a^{2}+\frac{12\!\cdots\!06}{62\!\cdots\!73}a-\frac{15\!\cdots\!99}{13\!\cdots\!59}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{1518148}$, which has order $24290368$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{71\!\cdots\!48}{42\!\cdots\!69}a^{15}-\frac{61\!\cdots\!58}{42\!\cdots\!69}a^{14}+\frac{47\!\cdots\!28}{42\!\cdots\!69}a^{13}-\frac{30\!\cdots\!50}{42\!\cdots\!69}a^{12}+\frac{12\!\cdots\!96}{42\!\cdots\!69}a^{11}-\frac{52\!\cdots\!72}{42\!\cdots\!69}a^{10}+\frac{14\!\cdots\!92}{42\!\cdots\!69}a^{9}-\frac{34\!\cdots\!15}{42\!\cdots\!69}a^{8}+\frac{79\!\cdots\!28}{42\!\cdots\!69}a^{7}-\frac{88\!\cdots\!32}{90\!\cdots\!27}a^{6}+\frac{18\!\cdots\!24}{42\!\cdots\!69}a^{5}+\frac{17\!\cdots\!78}{42\!\cdots\!69}a^{4}+\frac{18\!\cdots\!48}{42\!\cdots\!69}a^{3}+\frac{38\!\cdots\!68}{42\!\cdots\!69}a^{2}+\frac{72\!\cdots\!84}{42\!\cdots\!69}a+\frac{26\!\cdots\!97}{90\!\cdots\!27}$, $\frac{12\!\cdots\!44}{56\!\cdots\!97}a^{15}+\frac{11\!\cdots\!31}{56\!\cdots\!97}a^{14}-\frac{86\!\cdots\!56}{56\!\cdots\!97}a^{13}+\frac{58\!\cdots\!69}{56\!\cdots\!97}a^{12}-\frac{21\!\cdots\!32}{56\!\cdots\!97}a^{11}+\frac{10\!\cdots\!33}{56\!\cdots\!97}a^{10}-\frac{25\!\cdots\!02}{56\!\cdots\!97}a^{9}+\frac{69\!\cdots\!94}{56\!\cdots\!97}a^{8}-\frac{13\!\cdots\!88}{56\!\cdots\!97}a^{7}+\frac{25\!\cdots\!86}{11\!\cdots\!51}a^{6}-\frac{30\!\cdots\!06}{56\!\cdots\!97}a^{5}-\frac{17\!\cdots\!21}{56\!\cdots\!97}a^{4}-\frac{23\!\cdots\!50}{56\!\cdots\!97}a^{3}-\frac{36\!\cdots\!16}{56\!\cdots\!97}a^{2}-\frac{74\!\cdots\!28}{56\!\cdots\!97}a+\frac{49\!\cdots\!59}{11\!\cdots\!51}$, $\frac{19\!\cdots\!44}{56\!\cdots\!97}a^{15}-\frac{14\!\cdots\!85}{56\!\cdots\!97}a^{14}+\frac{12\!\cdots\!46}{56\!\cdots\!97}a^{13}-\frac{68\!\cdots\!18}{56\!\cdots\!97}a^{12}+\frac{32\!\cdots\!34}{56\!\cdots\!97}a^{11}-\frac{10\!\cdots\!91}{56\!\cdots\!97}a^{10}+\frac{39\!\cdots\!94}{56\!\cdots\!97}a^{9}-\frac{59\!\cdots\!00}{56\!\cdots\!97}a^{8}+\frac{23\!\cdots\!08}{56\!\cdots\!97}a^{7}+\frac{13\!\cdots\!52}{11\!\cdots\!51}a^{6}+\frac{67\!\cdots\!90}{56\!\cdots\!97}a^{5}+\frac{10\!\cdots\!77}{56\!\cdots\!97}a^{4}+\frac{88\!\cdots\!78}{56\!\cdots\!97}a^{3}+\frac{23\!\cdots\!86}{56\!\cdots\!97}a^{2}+\frac{42\!\cdots\!72}{56\!\cdots\!97}a+\frac{15\!\cdots\!49}{11\!\cdots\!51}$, $\frac{14\!\cdots\!51}{62\!\cdots\!73}a^{15}-\frac{12\!\cdots\!30}{62\!\cdots\!73}a^{14}+\frac{96\!\cdots\!91}{62\!\cdots\!73}a^{13}-\frac{58\!\cdots\!16}{62\!\cdots\!73}a^{12}+\frac{23\!\cdots\!25}{62\!\cdots\!73}a^{11}-\frac{31\!\cdots\!19}{20\!\cdots\!83}a^{10}+\frac{27\!\cdots\!74}{62\!\cdots\!73}a^{9}-\frac{58\!\cdots\!07}{62\!\cdots\!73}a^{8}+\frac{14\!\cdots\!14}{62\!\cdots\!73}a^{7}-\frac{10\!\cdots\!50}{13\!\cdots\!59}a^{6}+\frac{31\!\cdots\!56}{62\!\cdots\!73}a^{5}+\frac{26\!\cdots\!84}{62\!\cdots\!73}a^{4}+\frac{24\!\cdots\!52}{62\!\cdots\!73}a^{3}+\frac{36\!\cdots\!27}{62\!\cdots\!73}a^{2}+\frac{71\!\cdots\!12}{62\!\cdots\!73}a+\frac{17\!\cdots\!55}{13\!\cdots\!59}$, $\frac{10\!\cdots\!71}{62\!\cdots\!73}a^{15}-\frac{82\!\cdots\!37}{62\!\cdots\!73}a^{14}+\frac{70\!\cdots\!63}{62\!\cdots\!73}a^{13}-\frac{38\!\cdots\!45}{62\!\cdots\!73}a^{12}+\frac{17\!\cdots\!69}{62\!\cdots\!73}a^{11}-\frac{19\!\cdots\!36}{20\!\cdots\!83}a^{10}+\frac{20\!\cdots\!20}{62\!\cdots\!73}a^{9}-\frac{31\!\cdots\!16}{62\!\cdots\!73}a^{8}+\frac{10\!\cdots\!98}{62\!\cdots\!73}a^{7}+\frac{46\!\cdots\!80}{13\!\cdots\!59}a^{6}+\frac{25\!\cdots\!10}{62\!\cdots\!73}a^{5}+\frac{33\!\cdots\!21}{62\!\cdots\!73}a^{4}+\frac{24\!\cdots\!18}{62\!\cdots\!73}a^{3}+\frac{52\!\cdots\!39}{62\!\cdots\!73}a^{2}+\frac{95\!\cdots\!88}{62\!\cdots\!73}a+\frac{34\!\cdots\!17}{13\!\cdots\!59}$, $\frac{98\!\cdots\!19}{62\!\cdots\!73}a^{15}+\frac{96\!\cdots\!36}{62\!\cdots\!73}a^{14}-\frac{66\!\cdots\!46}{62\!\cdots\!73}a^{13}+\frac{49\!\cdots\!90}{62\!\cdots\!73}a^{12}-\frac{16\!\cdots\!92}{62\!\cdots\!73}a^{11}+\frac{28\!\cdots\!99}{20\!\cdots\!83}a^{10}-\frac{19\!\cdots\!82}{62\!\cdots\!73}a^{9}+\frac{65\!\cdots\!55}{62\!\cdots\!73}a^{8}-\frac{10\!\cdots\!12}{62\!\cdots\!73}a^{7}+\frac{31\!\cdots\!23}{13\!\cdots\!59}a^{6}-\frac{21\!\cdots\!42}{62\!\cdots\!73}a^{5}-\frac{18\!\cdots\!54}{62\!\cdots\!73}a^{4}-\frac{13\!\cdots\!24}{62\!\cdots\!73}a^{3}-\frac{11\!\cdots\!54}{62\!\cdots\!73}a^{2}-\frac{32\!\cdots\!26}{62\!\cdots\!73}a+\frac{52\!\cdots\!53}{13\!\cdots\!59}$, $\frac{13\!\cdots\!58}{62\!\cdots\!73}a^{15}-\frac{11\!\cdots\!35}{62\!\cdots\!73}a^{14}+\frac{91\!\cdots\!55}{62\!\cdots\!73}a^{13}-\frac{57\!\cdots\!32}{62\!\cdots\!73}a^{12}+\frac{22\!\cdots\!93}{62\!\cdots\!73}a^{11}-\frac{31\!\cdots\!17}{20\!\cdots\!83}a^{10}+\frac{26\!\cdots\!06}{62\!\cdots\!73}a^{9}-\frac{61\!\cdots\!54}{62\!\cdots\!73}a^{8}+\frac{14\!\cdots\!50}{62\!\cdots\!73}a^{7}-\frac{13\!\cdots\!71}{13\!\cdots\!59}a^{6}+\frac{35\!\cdots\!52}{62\!\cdots\!73}a^{5}+\frac{33\!\cdots\!72}{62\!\cdots\!73}a^{4}+\frac{33\!\cdots\!42}{62\!\cdots\!73}a^{3}+\frac{68\!\cdots\!97}{62\!\cdots\!73}a^{2}+\frac{13\!\cdots\!94}{62\!\cdots\!73}a+\frac{68\!\cdots\!79}{13\!\cdots\!59}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 32531.404863900207 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 32531.404863900207 \cdot 24290368}{2\cdot\sqrt{6896992068684009105950840251291271168}}\cr\approx \mathstrut & 0.365439931632864 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 664*x^14 - 3888*x^13 + 167028*x^12 - 646096*x^11 + 19823736*x^10 - 38554608*x^9 + 1128716344*x^8 - 144917456*x^7 + 29145288616*x^6 + 33165335168*x^5 + 345115392384*x^4 + 603518307936*x^3 + 2078152822832*x^2 + 2472153157008*x + 3733330807202)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^16 - 8*x^15 + 664*x^14 - 3888*x^13 + 167028*x^12 - 646096*x^11 + 19823736*x^10 - 38554608*x^9 + 1128716344*x^8 - 144917456*x^7 + 29145288616*x^6 + 33165335168*x^5 + 345115392384*x^4 + 603518307936*x^3 + 2078152822832*x^2 + 2472153157008*x + 3733330807202, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 8*x^15 + 664*x^14 - 3888*x^13 + 167028*x^12 - 646096*x^11 + 19823736*x^10 - 38554608*x^9 + 1128716344*x^8 - 144917456*x^7 + 29145288616*x^6 + 33165335168*x^5 + 345115392384*x^4 + 603518307936*x^3 + 2078152822832*x^2 + 2472153157008*x + 3733330807202);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 664*x^14 - 3888*x^13 + 167028*x^12 - 646096*x^11 + 19823736*x^10 - 38554608*x^9 + 1128716344*x^8 - 144917456*x^7 + 29145288616*x^6 + 33165335168*x^5 + 345115392384*x^4 + 603518307936*x^3 + 2078152822832*x^2 + 2472153157008*x + 3733330807202);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_2^5.C_2\wr C_4$ (as 16T1354):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 2048 |
| The 59 conjugacy class representatives for $C_2^5.C_2\wr C_4$ |
| Character table for $C_2^5.C_2\wr C_4$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.7583301632.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $16$ | $16$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{6}$ | $16$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ | $16$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | $16$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | $16$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | $16$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $16$ | $1$ | $62$ | |||
|
\(113\)
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 113.2.1.2 | $x^{2} + 339$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 113.4.0.1 | $x^{4} + 62 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 113.4.0.1 | $x^{4} + 62 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 113.4.2.1 | $x^{4} + 18960 x^{3} + 90817911 x^{2} + 8982404280 x + 374946100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(1009\)
| $\Q_{1009}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1009}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1009}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1009}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1009}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1009}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |