Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 9*x^18 - 18*x^17 + 33*x^16 - 50*x^15 + 73*x^14 - 93*x^13 + 118*x^12 - 128*x^11 + 135*x^10 - 112*x^9 + 88*x^8 - 48*x^7 + 27*x^6 - 10*x^5 + 10*x^4 - 6*x^3 + 6*x^2 - 2*x + 1)
gp: K = bnfinit(y^20 - 3*y^19 + 9*y^18 - 18*y^17 + 33*y^16 - 50*y^15 + 73*y^14 - 93*y^13 + 118*y^12 - 128*y^11 + 135*y^10 - 112*y^9 + 88*y^8 - 48*y^7 + 27*y^6 - 10*y^5 + 10*y^4 - 6*y^3 + 6*y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 3*x^19 + 9*x^18 - 18*x^17 + 33*x^16 - 50*x^15 + 73*x^14 - 93*x^13 + 118*x^12 - 128*x^11 + 135*x^10 - 112*x^9 + 88*x^8 - 48*x^7 + 27*x^6 - 10*x^5 + 10*x^4 - 6*x^3 + 6*x^2 - 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 3*x^19 + 9*x^18 - 18*x^17 + 33*x^16 - 50*x^15 + 73*x^14 - 93*x^13 + 118*x^12 - 128*x^11 + 135*x^10 - 112*x^9 + 88*x^8 - 48*x^7 + 27*x^6 - 10*x^5 + 10*x^4 - 6*x^3 + 6*x^2 - 2*x + 1)
\( x^{20} - 3 x^{19} + 9 x^{18} - 18 x^{17} + 33 x^{16} - 50 x^{15} + 73 x^{14} - 93 x^{13} + 118 x^{12} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2615059431182861328125\)
\(\medspace = 5^{15}\cdot 199^{2}\cdot 1471^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.77\) | 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: | $5^{3/4}199^{1/2}1471^{1/2}\approx 1809.0908531435562$ | ||
| Ramified primes: |
\(5\), \(199\), \(1471\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{287389}a^{19}+\frac{98926}{287389}a^{18}-\frac{94743}{287389}a^{17}+\frac{74581}{287389}a^{16}+\frac{85985}{287389}a^{15}-\frac{16996}{287389}a^{14}+\frac{115828}{287389}a^{13}-\frac{26089}{287389}a^{12}+\frac{82046}{287389}a^{11}+\frac{1079}{287389}a^{10}+\frac{123207}{287389}a^{9}+\frac{2923}{287389}a^{8}+\frac{56221}{287389}a^{7}+\frac{47944}{287389}a^{6}-\frac{16053}{287389}a^{5}+\frac{4367}{287389}a^{4}+\frac{77286}{287389}a^{3}+\frac{129732}{287389}a^{2}+\frac{39072}{287389}a-\frac{28164}{287389}$
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
Trivial group, which has order $1$
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: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{674473}{287389} a^{19} - \frac{1912466}{287389} a^{18} + \frac{5383880}{287389} a^{17} - \frac{10321417}{287389} a^{16} + \frac{17853601}{287389} a^{15} - \frac{25835686}{287389} a^{14} + \frac{36393454}{287389} a^{13} - \frac{44617700}{287389} a^{12} + \frac{55088940}{287389} a^{11} - \frac{56815603}{287389} a^{10} + \frac{56156860}{287389} a^{9} - \frac{41100588}{287389} a^{8} + \frac{27306883}{287389} a^{7} - \frac{10997550}{287389} a^{6} + \frac{4951119}{287389} a^{5} - \frac{2612771}{287389} a^{4} + \frac{4726904}{287389} a^{3} - \frac{2609317}{287389} a^{2} + \frac{1449479}{287389} a - \frac{19450}{287389} \)
(order $10$)
| 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: |
$\frac{1529931}{287389}a^{19}-\frac{4037522}{287389}a^{18}+\frac{11632757}{287389}a^{17}-\frac{21549268}{287389}a^{16}+\frac{37599800}{287389}a^{15}-\frac{53492299}{287389}a^{14}+\frac{76062940}{287389}a^{13}-\frac{91738296}{287389}a^{12}+\frac{114769173}{287389}a^{11}-\frac{115209856}{287389}a^{10}+\frac{116156162}{287389}a^{9}-\frac{80828225}{287389}a^{8}+\frac{57063018}{287389}a^{7}-\frac{19974614}{287389}a^{6}+\frac{11777157}{287389}a^{5}-\frac{4034241}{287389}a^{4}+\frac{9837888}{287389}a^{3}-\frac{4204358}{287389}a^{2}+\frac{3425922}{287389}a-\frac{169136}{287389}$, $\frac{19450}{287389}a^{19}+\frac{616123}{287389}a^{18}-\frac{1737416}{287389}a^{17}+\frac{5033780}{287389}a^{16}-\frac{9679567}{287389}a^{15}+\frac{16881101}{287389}a^{14}-\frac{24415836}{287389}a^{13}+\frac{34584604}{287389}a^{12}-\frac{42322600}{287389}a^{11}+\frac{52599340}{287389}a^{10}-\frac{54189853}{287389}a^{9}+\frac{53978460}{287389}a^{8}-\frac{39388988}{287389}a^{7}+\frac{26373283}{287389}a^{6}-\frac{10472400}{287389}a^{5}+\frac{4756619}{287389}a^{4}-\frac{2418271}{287389}a^{3}+\frac{4610204}{287389}a^{2}-\frac{2492617}{287389}a+\frac{1410579}{287389}$, $\frac{596514}{287389}a^{19}-\frac{2288074}{287389}a^{18}+\frac{6418284}{287389}a^{17}-\frac{13863405}{287389}a^{16}+\frac{24794747}{287389}a^{15}-\frac{38640317}{287389}a^{14}+\frac{55001067}{287389}a^{13}-\frac{71899257}{287389}a^{12}+\frac{88331534}{287389}a^{11}-\frac{99549348}{287389}a^{10}+\frac{100148022}{287389}a^{9}-\frac{87057508}{287389}a^{8}+\frac{60968096}{287389}a^{7}-\frac{35310577}{287389}a^{6}+\frac{13756910}{287389}a^{5}-\frac{6814594}{287389}a^{4}+\frac{4985404}{287389}a^{3}-\frac{6041285}{287389}a^{2}+\frac{3195776}{287389}a-\frac{1758468}{287389}$, $\frac{1226}{4871}a^{19}-\frac{28979}{4871}a^{18}+\frac{76884}{4871}a^{17}-\frac{211559}{4871}a^{16}+\frac{389108}{4871}a^{15}-\frac{666285}{4871}a^{14}+\frac{945839}{4871}a^{13}-\frac{1327040}{4871}a^{12}+\frac{1599934}{4871}a^{11}-\frac{1974813}{4871}a^{10}+\frac{1974827}{4871}a^{9}-\frac{1949858}{4871}a^{8}+\frac{1341821}{4871}a^{7}-\frac{909890}{4871}a^{6}+\frac{309606}{4871}a^{5}-\frac{169772}{4871}a^{4}+\frac{75009}{4871}a^{3}-\frac{162074}{4871}a^{2}+\frac{69052}{4871}a-\frac{52126}{4871}$, $\frac{2348837}{287389}a^{19}-\frac{5602554}{287389}a^{18}+\frac{16165986}{287389}a^{17}-\frac{28378942}{287389}a^{16}+\frac{48686713}{287389}a^{15}-\frac{66589299}{287389}a^{14}+\frac{94247943}{287389}a^{13}-\frac{109209399}{287389}a^{12}+\frac{137335048}{287389}a^{11}-\frac{129126129}{287389}a^{10}+\frac{128947034}{287389}a^{9}-\frac{75655966}{287389}a^{8}+\frac{51886442}{287389}a^{7}-\frac{7348469}{287389}a^{6}+\frac{8178109}{287389}a^{5}-\frac{2128732}{287389}a^{4}+\frac{12725758}{287389}a^{3}-\frac{2396295}{287389}a^{2}+\frac{2979450}{287389}a+\frac{1141253}{287389}$, $\frac{122583}{287389}a^{19}+\frac{267003}{287389}a^{18}-\frac{779068}{287389}a^{17}+\frac{3105134}{287389}a^{16}-\frac{6302267}{287389}a^{15}+\frac{11932531}{287389}a^{14}-\frac{17440550}{287389}a^{13}+\frac{25861915}{287389}a^{12}-\frac{31633416}{287389}a^{11}+\frac{41164744}{287389}a^{10}-\frac{42891397}{287389}a^{9}+\frac{45343488}{287389}a^{8}-\frac{33473890}{287389}a^{7}+\frac{23867589}{287389}a^{6}-\frac{8981475}{287389}a^{5}+\frac{3937700}{287389}a^{4}-\frac{978203}{287389}a^{3}+\frac{3716109}{287389}a^{2}-\frac{2073821}{287389}a+\frac{1413390}{287389}$, $\frac{93461}{287389}a^{19}-\frac{443411}{287389}a^{18}+\frac{1116512}{287389}a^{17}-\frac{2504466}{287389}a^{16}+\frac{4296313}{287389}a^{15}-\frac{6674100}{287389}a^{14}+\frac{9228304}{287389}a^{13}-\frac{12166091}{287389}a^{12}+\frac{14357358}{287389}a^{11}-\frac{16410293}{287389}a^{10}+\frac{15465981}{287389}a^{9}-\frac{13340330}{287389}a^{8}+\frac{7897297}{287389}a^{7}-\frac{4385939}{287389}a^{6}+\frac{703314}{287389}a^{5}-\frac{235582}{287389}a^{4}+\frac{279109}{287389}a^{3}-\frac{634236}{287389}a^{2}+\frac{143558}{287389}a-\frac{327142}{287389}$, $\frac{19169}{287389}a^{19}+\frac{407261}{287389}a^{18}-\frac{979643}{287389}a^{17}+\frac{3044193}{287389}a^{16}-\frac{5389841}{287389}a^{15}+\frac{9586639}{287389}a^{14}-\frac{13280376}{287389}a^{13}+\frac{19211882}{287389}a^{12}-\frac{22556565}{287389}a^{11}+\frac{29017632}{287389}a^{10}-\frac{27884552}{287389}a^{9}+\frac{29016421}{287389}a^{8}-\frac{18401297}{287389}a^{7}+\frac{14050575}{287389}a^{6}-\frac{3662395}{287389}a^{5}+\frac{3242103}{287389}a^{4}-\frac{569739}{287389}a^{3}+\frac{2354803}{287389}a^{2}-\frac{539344}{287389}a+\frac{990382}{287389}$, $\frac{659985}{287389}a^{19}-\frac{2805578}{287389}a^{18}+\frac{7749712}{287389}a^{17}-\frac{17165641}{287389}a^{16}+\frac{30579352}{287389}a^{15}-\frac{48018964}{287389}a^{14}+\frac{67954551}{287389}a^{13}-\frac{89389487}{287389}a^{12}+\frac{109076528}{287389}a^{11}-\frac{124178175}{287389}a^{10}+\frac{123643338}{287389}a^{9}-\frac{108739244}{287389}a^{8}+\frac{74081868}{287389}a^{7}-\frac{42891188}{287389}a^{6}+\frac{15662675}{287389}a^{5}-\frac{7829289}{287389}a^{4}+\frac{6399214}{287389}a^{3}-\frac{7815475}{287389}a^{2}+\frac{4117174}{287389}a-\frac{2083521}{287389}$
| 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: | \( 906.492535524 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 906.492535524 \cdot 1}{10\cdot\sqrt{2615059431182861328125}}\cr\approx \mathstrut & 0.169989514761 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 9*x^18 - 18*x^17 + 33*x^16 - 50*x^15 + 73*x^14 - 93*x^13 + 118*x^12 - 128*x^11 + 135*x^10 - 112*x^9 + 88*x^8 - 48*x^7 + 27*x^6 - 10*x^5 + 10*x^4 - 6*x^3 + 6*x^2 - 2*x + 1)
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^20 - 3*x^19 + 9*x^18 - 18*x^17 + 33*x^16 - 50*x^15 + 73*x^14 - 93*x^13 + 118*x^12 - 128*x^11 + 135*x^10 - 112*x^9 + 88*x^8 - 48*x^7 + 27*x^6 - 10*x^5 + 10*x^4 - 6*x^3 + 6*x^2 - 2*x + 1, 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^20 - 3*x^19 + 9*x^18 - 18*x^17 + 33*x^16 - 50*x^15 + 73*x^14 - 93*x^13 + 118*x^12 - 128*x^11 + 135*x^10 - 112*x^9 + 88*x^8 - 48*x^7 + 27*x^6 - 10*x^5 + 10*x^4 - 6*x^3 + 6*x^2 - 2*x + 1);
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^20 - 3*x^19 + 9*x^18 - 18*x^17 + 33*x^16 - 50*x^15 + 73*x^14 - 93*x^13 + 118*x^12 - 128*x^11 + 135*x^10 - 112*x^9 + 88*x^8 - 48*x^7 + 27*x^6 - 10*x^5 + 10*x^4 - 6*x^3 + 6*x^2 - 2*x + 1);
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
$S_5^2:C_4$ (as 20T654):
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 non-solvable group of order 57600 |
| The 70 conjugacy class representatives for $S_5^2:C_4$ |
| Character table for $S_5^2:C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.2.914778125.1 |
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 20 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 40 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 | $20$ | $20$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | $20$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ | $20$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.12.9.2 | $x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
|
\(199\)
| 199.2.0.1 | $x^{2} + 193 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 199.2.0.1 | $x^{2} + 193 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 199.2.0.1 | $x^{2} + 193 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 199.2.0.1 | $x^{2} + 193 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 199.2.0.1 | $x^{2} + 193 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 199.4.2.1 | $x^{4} + 386 x^{3} + 37653 x^{2} + 77972 x + 7450967$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 199.6.0.1 | $x^{6} + 90 x^{3} + 58 x^{2} + 79 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(1471\)
| $\Q_{1471}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1471}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1471}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1471}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1471}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1471}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1471}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1471}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |