Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 27*x^13 - 4*x^12 + 252*x^11 + 60*x^10 - 976*x^9 - 288*x^8 + 1473*x^7 + 384*x^6 - 765*x^5 - 168*x^4 + 150*x^3 + 27*x^2 - 9*x - 1)
gp: K = bnfinit(y^15 - 27*y^13 - 4*y^12 + 252*y^11 + 60*y^10 - 976*y^9 - 288*y^8 + 1473*y^7 + 384*y^6 - 765*y^5 - 168*y^4 + 150*y^3 + 27*y^2 - 9*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 27*x^13 - 4*x^12 + 252*x^11 + 60*x^10 - 976*x^9 - 288*x^8 + 1473*x^7 + 384*x^6 - 765*x^5 - 168*x^4 + 150*x^3 + 27*x^2 - 9*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 27*x^13 - 4*x^12 + 252*x^11 + 60*x^10 - 976*x^9 - 288*x^8 + 1473*x^7 + 384*x^6 - 765*x^5 - 168*x^4 + 150*x^3 + 27*x^2 - 9*x - 1)
\( x^{15} - 27 x^{13} - 4 x^{12} + 252 x^{11} + 60 x^{10} - 976 x^{9} - 288 x^{8} + 1473 x^{7} + 384 x^{6} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[15, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(10943023107606534329121\)
\(\medspace = 3^{20}\cdot 11^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(29.46\) | 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: | $3^{4/3}11^{4/5}\approx 29.46292234405376$ | ||
| Ramified primes: |
\(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $15$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(99=3^{2}\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{99}(64,·)$, $\chi_{99}(1,·)$, $\chi_{99}(34,·)$, $\chi_{99}(67,·)$, $\chi_{99}(4,·)$, $\chi_{99}(37,·)$, $\chi_{99}(70,·)$, $\chi_{99}(97,·)$, $\chi_{99}(16,·)$, $\chi_{99}(49,·)$, $\chi_{99}(82,·)$, $\chi_{99}(25,·)$, $\chi_{99}(58,·)$, $\chi_{99}(91,·)$, $\chi_{99}(31,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{164276239249}a^{14}+\frac{32330953608}{164276239249}a^{13}+\frac{74618147066}{164276239249}a^{12}-\frac{7544382288}{164276239249}a^{11}-\frac{50914705859}{164276239249}a^{10}+\frac{3571539703}{164276239249}a^{9}-\frac{62434792324}{164276239249}a^{8}-\frac{80812000837}{164276239249}a^{7}-\frac{13866789831}{164276239249}a^{6}-\frac{48159141661}{164276239249}a^{5}+\frac{60633542357}{164276239249}a^{4}-\frac{855515036}{164276239249}a^{3}-\frac{76391229334}{164276239249}a^{2}+\frac{263409277}{164276239249}a+\frac{11630783081}{164276239249}$
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 $1$ (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: | $14$ | 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: |
$\frac{376330653028}{164276239249}a^{14}-\frac{264234824073}{164276239249}a^{13}-\frac{10099735865109}{164276239249}a^{12}+\frac{5573747051824}{164276239249}a^{11}+\frac{94248126319446}{164276239249}a^{10}-\frac{42770903224203}{164276239249}a^{9}-\frac{367728228237766}{164276239249}a^{8}+\frac{139521583807254}{164276239249}a^{7}+\frac{569463871754778}{164276239249}a^{6}-\frac{210708089001740}{164276239249}a^{5}-\frac{291841072140321}{164276239249}a^{4}+\frac{89399998079502}{164276239249}a^{3}+\frac{47901544587303}{164276239249}a^{2}-\frac{9471173638311}{164276239249}a-\frac{1601326448183}{164276239249}$, $\frac{322512271017}{164276239249}a^{14}-\frac{75644052878}{164276239249}a^{13}-\frac{8639934299064}{164276239249}a^{12}+\frac{740643695655}{164276239249}a^{11}+\frac{79757341304444}{164276239249}a^{10}+\frac{329073971337}{164276239249}a^{9}-\frac{302553803789352}{164276239249}a^{8}-\frac{17900422836336}{164276239249}a^{7}+\frac{433572336372033}{164276239249}a^{6}+\frac{4470215964924}{164276239249}a^{5}-\frac{186301503342354}{164276239249}a^{4}+\frac{10503280180593}{164276239249}a^{3}+\frac{23967582145383}{164276239249}a^{2}-\frac{2560031767662}{164276239249}a-\frac{566911470079}{164276239249}$, $\frac{411342166449}{164276239249}a^{14}-\frac{211540848828}{164276239249}a^{13}-\frac{11014492076914}{164276239249}a^{12}+\frac{4019631372684}{164276239249}a^{11}+\frac{102048251888454}{164276239249}a^{10}-\frac{27747596670390}{164276239249}a^{9}-\frac{391420151120322}{164276239249}a^{8}+\frac{81972070684713}{164276239249}a^{7}+\frac{579729426648720}{164276239249}a^{6}-\frac{136064345279814}{164276239249}a^{5}-\frac{267414456041310}{164276239249}a^{4}+\frac{63645080648632}{164276239249}a^{3}+\frac{38518648026174}{164276239249}a^{2}-\frac{7412210966880}{164276239249}a-\frac{898464267285}{164276239249}$, $\frac{12762505737}{164276239249}a^{14}+\frac{116316639441}{164276239249}a^{13}-\frac{328329840878}{164276239249}a^{12}-\frac{3162740573502}{164276239249}a^{11}+\frac{2320190824137}{164276239249}a^{10}+\frac{29240124522596}{164276239249}a^{9}-\frac{1718657348976}{164276239249}a^{8}-\frac{109174521913335}{164276239249}a^{7}-\frac{26802790136704}{164276239249}a^{6}+\frac{145678759197039}{164276239249}a^{5}+\frac{43312423035948}{164276239249}a^{4}-\frac{51540058502331}{164276239249}a^{3}-\frac{11177403685335}{164276239249}a^{2}+\frac{4742259305952}{164276239249}a+\frac{327741632820}{164276239249}$, $\frac{264234824073}{164276239249}a^{14}-\frac{61191766647}{164276239249}a^{13}-\frac{7079069663936}{164276239249}a^{12}+\frac{587198243610}{164276239249}a^{11}+\frac{65350742405883}{164276239249}a^{10}+\frac{429510882438}{164276239249}a^{9}-\frac{247904811879318}{164276239249}a^{8}-\frac{15128819844534}{164276239249}a^{7}+\frac{355219059764492}{164276239249}a^{6}+\frac{3948122573901}{164276239249}a^{5}-\frac{152623547788206}{164276239249}a^{4}+\frac{8548053366897}{164276239249}a^{3}+\frac{19632101270067}{164276239249}a^{2}-\frac{1621373189820}{164276239249}a-\frac{212054413779}{164276239249}$, $\frac{264234824073}{164276239249}a^{14}-\frac{61191766647}{164276239249}a^{13}-\frac{7079069663936}{164276239249}a^{12}+\frac{587198243610}{164276239249}a^{11}+\frac{65350742405883}{164276239249}a^{10}+\frac{429510882438}{164276239249}a^{9}-\frac{247904811879318}{164276239249}a^{8}-\frac{15128819844534}{164276239249}a^{7}+\frac{355219059764492}{164276239249}a^{6}+\frac{3948122573901}{164276239249}a^{5}-\frac{152623547788206}{164276239249}a^{4}+\frac{8548053366897}{164276239249}a^{3}+\frac{19632101270067}{164276239249}a^{2}-\frac{1621373189820}{164276239249}a-\frac{376330653028}{164276239249}$, $\frac{154206270031}{164276239249}a^{14}-\frac{128408041174}{164276239249}a^{13}-\frac{4080975970923}{164276239249}a^{12}+\frac{2820732419358}{164276239249}a^{11}+\frac{37161647641349}{164276239249}a^{10}-\frac{22648590550179}{164276239249}a^{9}-\frac{137759544898184}{164276239249}a^{8}+\frac{78559654711091}{164276239249}a^{7}+\frac{186163953295627}{164276239249}a^{6}-\frac{125512486783155}{164276239249}a^{5}-\frac{53780444299022}{164276239249}a^{4}+\frac{61330729665941}{164276239249}a^{3}-\frac{5898588981172}{164276239249}a^{2}-\frac{7937197309813}{164276239249}a+\frac{1957493944456}{164276239249}$, $\frac{640565477101}{164276239249}a^{14}-\frac{325426590720}{164276239249}a^{13}-\frac{17178805529045}{164276239249}a^{12}+\frac{6160945295434}{164276239249}a^{11}+\frac{159598868725329}{164276239249}a^{10}-\frac{42341392341765}{164276239249}a^{9}-\frac{615633040117084}{164276239249}a^{8}+\frac{124392763962720}{164276239249}a^{7}+\frac{924682931519270}{164276239249}a^{6}-\frac{206759966427839}{164276239249}a^{5}-\frac{444464619928527}{164276239249}a^{4}+\frac{97948051446399}{164276239249}a^{3}+\frac{67533645857370}{164276239249}a^{2}-\frac{11092546828131}{164276239249}a-\frac{1977657101211}{164276239249}$, $\frac{330536140}{1507121461}a^{14}+\frac{451104385}{1507121461}a^{13}-\frac{8939800589}{1507121461}a^{12}-\frac{13350270113}{1507121461}a^{11}+\frac{81896001410}{1507121461}a^{10}+\frac{129508083098}{1507121461}a^{9}-\frac{299777306206}{1507121461}a^{8}-\frac{499051484244}{1507121461}a^{7}+\frac{378760840155}{1507121461}a^{6}+\frac{656487237304}{1507121461}a^{5}-\frac{135707785479}{1507121461}a^{4}-\frac{218623066585}{1507121461}a^{3}+\frac{23311685790}{1507121461}a^{2}+\frac{16005853731}{1507121461}a-\frac{3682799320}{1507121461}$, $\frac{116151399073}{164276239249}a^{14}-\frac{197697840174}{164276239249}a^{13}-\frac{3130011144669}{164276239249}a^{12}+\frac{4833570237772}{164276239249}a^{11}+\frac{29892757322751}{164276239249}a^{10}-\frac{41811920368299}{164276239249}a^{9}-\frac{123422981848519}{164276239249}a^{8}+\frac{150098668364052}{164276239249}a^{7}+\frac{218694713053788}{164276239249}a^{6}-\frac{211873514832101}{164276239249}a^{5}-\frac{140976812824731}{164276239249}a^{4}+\frac{84325590124556}{164276239249}a^{3}+\frac{28465396049634}{164276239249}a^{2}-\frac{8791761088632}{164276239249}a-\frac{1226751444107}{164276239249}$, $\frac{292270098161}{164276239249}a^{14}-\frac{84143745533}{164276239249}a^{13}-\frac{7848355796861}{164276239249}a^{12}+\frac{1090274628733}{164276239249}a^{11}+\frac{72833705158046}{164276239249}a^{10}-\frac{3502624298576}{164276239249}a^{9}-\frac{279553281276906}{164276239249}a^{8}-\frac{2617175424007}{164276239249}a^{7}+\frac{413214035053101}{164276239249}a^{6}-\frac{11845509573861}{164276239249}a^{5}-\frac{193467073348398}{164276239249}a^{4}+\frac{12834918136824}{164276239249}a^{3}+\frac{27392540203071}{164276239249}a^{2}-\frac{1852481341690}{164276239249}a-\frac{489932787919}{164276239249}$, $\frac{39992328004}{164276239249}a^{14}+\frac{54853901655}{164276239249}a^{13}-\frac{1087659883976}{164276239249}a^{12}-\frac{1629702449832}{164276239249}a^{11}+\frac{10074930052770}{164276239249}a^{10}+\frac{15944319991099}{164276239249}a^{9}-\frac{37856101859186}{164276239249}a^{8}-\frac{62603110080682}{164276239249}a^{7}+\frac{52063164034107}{164276239249}a^{6}+\frac{87087442140829}{164276239249}a^{5}-\frac{25566063040429}{164276239249}a^{4}-\frac{35516067827051}{164276239249}a^{3}+\frac{7063639244037}{164276239249}a^{2}+\frac{3956589533414}{164276239249}a-\frac{801136868929}{164276239249}$, $\frac{247613191937}{164276239249}a^{14}-\frac{59735034653}{164276239249}a^{13}-\frac{6646675879732}{164276239249}a^{12}+\frac{615354397079}{164276239249}a^{11}+\frac{61596028556955}{164276239249}a^{10}-\frac{163569478331}{164276239249}a^{9}-\frac{235652882213627}{164276239249}a^{8}-\frac{12446424896409}{164276239249}a^{7}+\frac{345650006330807}{164276239249}a^{6}+\frac{2968131764321}{164276239249}a^{5}-\frac{160985336371908}{164276239249}a^{4}+\frac{7118682524408}{164276239249}a^{3}+\frac{25947453322985}{164276239249}a^{2}-\frac{1260821457255}{164276239249}a-\frac{1094985144409}{164276239249}$, $\frac{565169032303}{164276239249}a^{14}+\frac{42798513928}{164276239249}a^{13}-\frac{15160755188855}{164276239249}a^{12}-\frac{3411844224292}{164276239249}a^{11}+\frac{139603599912649}{164276239249}a^{10}+\frac{44181892072025}{164276239249}a^{9}-\frac{524695717822665}{164276239249}a^{8}-\frac{197599344958266}{164276239249}a^{7}+\frac{729012621875702}{164276239249}a^{6}+\frac{248713397764864}{164276239249}a^{5}-\frac{290567008686901}{164276239249}a^{4}-\frac{89783791011393}{164276239249}a^{3}+\frac{29568939555247}{164276239249}a^{2}+\frac{9640706543019}{164276239249}a+\frac{687665361962}{164276239249}$
| 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: | \( 967645.576239 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 967645.576239 \cdot 1}{2\cdot\sqrt{10943023107606534329121}}\cr\approx \mathstrut & 0.151554067476 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 27*x^13 - 4*x^12 + 252*x^11 + 60*x^10 - 976*x^9 - 288*x^8 + 1473*x^7 + 384*x^6 - 765*x^5 - 168*x^4 + 150*x^3 + 27*x^2 - 9*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^15 - 27*x^13 - 4*x^12 + 252*x^11 + 60*x^10 - 976*x^9 - 288*x^8 + 1473*x^7 + 384*x^6 - 765*x^5 - 168*x^4 + 150*x^3 + 27*x^2 - 9*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^15 - 27*x^13 - 4*x^12 + 252*x^11 + 60*x^10 - 976*x^9 - 288*x^8 + 1473*x^7 + 384*x^6 - 765*x^5 - 168*x^4 + 150*x^3 + 27*x^2 - 9*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^15 - 27*x^13 - 4*x^12 + 252*x^11 + 60*x^10 - 976*x^9 - 288*x^8 + 1473*x^7 + 384*x^6 - 765*x^5 - 168*x^4 + 150*x^3 + 27*x^2 - 9*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
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 cyclic group of order 15 |
| The 15 conjugacy class representatives for $C_{15}$ |
| Character table for $C_{15}$ |
Intermediate fields
| \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{11})^+\) |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $15$ | R | $15$ | $15$ | R | $15$ | ${\href{/padicField/17.5.0.1}{5} }^{3}$ | ${\href{/padicField/19.5.0.1}{5} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{5}$ | $15$ | $15$ | ${\href{/padicField/37.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/43.3.0.1}{3} }^{5}$ | $15$ | ${\href{/padicField/53.5.0.1}{5} }^{3}$ | $15$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.15.20.65 | $x^{15} + 30 x^{14} + 360 x^{13} + 2175 x^{12} + 6840 x^{11} + 11016 x^{10} + 13050 x^{9} + 21060 x^{8} + 9720 x^{7} + 24084 x^{6} + 55728 x^{5} + 167184 x^{4} + 79137 x^{3} + 474822 x^{2} + 138024$ | $3$ | $5$ | $20$ | $C_{15}$ | $[2]^{5}$ |
|
\(11\)
| 11.15.12.1 | $x^{15} + 10 x^{13} + 45 x^{12} + 40 x^{11} + 393 x^{10} + 890 x^{9} + 750 x^{8} - 3970 x^{7} + 9610 x^{6} + 13085 x^{5} + 151045 x^{4} + 26525 x^{3} + 116170 x^{2} - 53795 x + 67662$ | $5$ | $3$ | $12$ | $C_{15}$ | $[\ ]_{5}^{3}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.11.5t1.a.a | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.99.15t1.a.a | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ |
| * | 1.99.15t1.a.b | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ |
| * | 1.11.5t1.a.b | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.99.15t1.a.c | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ |
| * | 1.99.15t1.a.d | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ |
| * | 1.11.5t1.a.c | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.99.15t1.a.e | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ |
| * | 1.99.15t1.a.f | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ |
| * | 1.11.5t1.a.d | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.99.15t1.a.g | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ |
| * | 1.99.15t1.a.h | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.