Properties

Label 15.15.109...121.1
Degree $15$
Signature $[15, 0]$
Discriminant $1.094\times 10^{22}$
Root discriminant \(29.46\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{15}$ (as 15T1)

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Show commands: SageMath / Pari/GP / Magma

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(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)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -9, 27, 150, -168, -765, 384, 1473, -288, -976, 60, 252, -4, -27, 0, 1]);
 

\( x^{15} - 27 x^{13} - 4 x^{12} + 252 x^{11} + 60 x^{10} - 976 x^{9} - 288 x^{8} + 1473 x^{7} + 384 x^{6} - 765 x^{5} - 168 x^{4} + 150 x^{3} + 27 x^{2} - 9 x - 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $15$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[15, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:   \(10943023107606534329121\) \(\medspace = 3^{20}\cdot 11^{12}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(29.46\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:   \(3\), \(11\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$\card{ \Gal(K/\Q) }$:  $15$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{164276239249}a^{14}+\frac{32330953608}{164276239249}a^{13}+\frac{74618147066}{164276239249}a^{12}-\frac{7544382288}{164276239249}a^{11}-\frac{50914705859}{164276239249}a^{10}+\frac{3571539703}{164276239249}a^{9}-\frac{62434792324}{164276239249}a^{8}-\frac{80812000837}{164276239249}a^{7}-\frac{13866789831}{164276239249}a^{6}-\frac{48159141661}{164276239249}a^{5}+\frac{60633542357}{164276239249}a^{4}-\frac{855515036}{164276239249}a^{3}-\frac{76391229334}{164276239249}a^{2}+\frac{263409277}{164276239249}a+\frac{11630783081}{164276239249}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

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);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:   $\frac{376330653028}{164276239249}a^{14}-\frac{264234824073}{164276239249}a^{13}-\frac{10099735865109}{164276239249}a^{12}+\frac{5573747051824}{164276239249}a^{11}+\frac{94248126319446}{164276239249}a^{10}-\frac{42770903224203}{164276239249}a^{9}-\frac{367728228237766}{164276239249}a^{8}+\frac{139521583807254}{164276239249}a^{7}+\frac{569463871754778}{164276239249}a^{6}-\frac{210708089001740}{164276239249}a^{5}-\frac{291841072140321}{164276239249}a^{4}+\frac{89399998079502}{164276239249}a^{3}+\frac{47901544587303}{164276239249}a^{2}-\frac{9471173638311}{164276239249}a-\frac{1601326448183}{164276239249}$, $\frac{322512271017}{164276239249}a^{14}-\frac{75644052878}{164276239249}a^{13}-\frac{8639934299064}{164276239249}a^{12}+\frac{740643695655}{164276239249}a^{11}+\frac{79757341304444}{164276239249}a^{10}+\frac{329073971337}{164276239249}a^{9}-\frac{302553803789352}{164276239249}a^{8}-\frac{17900422836336}{164276239249}a^{7}+\frac{433572336372033}{164276239249}a^{6}+\frac{4470215964924}{164276239249}a^{5}-\frac{186301503342354}{164276239249}a^{4}+\frac{10503280180593}{164276239249}a^{3}+\frac{23967582145383}{164276239249}a^{2}-\frac{2560031767662}{164276239249}a-\frac{566911470079}{164276239249}$, $\frac{411342166449}{164276239249}a^{14}-\frac{211540848828}{164276239249}a^{13}-\frac{11014492076914}{164276239249}a^{12}+\frac{4019631372684}{164276239249}a^{11}+\frac{102048251888454}{164276239249}a^{10}-\frac{27747596670390}{164276239249}a^{9}-\frac{391420151120322}{164276239249}a^{8}+\frac{81972070684713}{164276239249}a^{7}+\frac{579729426648720}{164276239249}a^{6}-\frac{136064345279814}{164276239249}a^{5}-\frac{267414456041310}{164276239249}a^{4}+\frac{63645080648632}{164276239249}a^{3}+\frac{38518648026174}{164276239249}a^{2}-\frac{7412210966880}{164276239249}a-\frac{898464267285}{164276239249}$, $\frac{12762505737}{164276239249}a^{14}+\frac{116316639441}{164276239249}a^{13}-\frac{328329840878}{164276239249}a^{12}-\frac{3162740573502}{164276239249}a^{11}+\frac{2320190824137}{164276239249}a^{10}+\frac{29240124522596}{164276239249}a^{9}-\frac{1718657348976}{164276239249}a^{8}-\frac{109174521913335}{164276239249}a^{7}-\frac{26802790136704}{164276239249}a^{6}+\frac{145678759197039}{164276239249}a^{5}+\frac{43312423035948}{164276239249}a^{4}-\frac{51540058502331}{164276239249}a^{3}-\frac{11177403685335}{164276239249}a^{2}+\frac{4742259305952}{164276239249}a+\frac{327741632820}{164276239249}$, $\frac{264234824073}{164276239249}a^{14}-\frac{61191766647}{164276239249}a^{13}-\frac{7079069663936}{164276239249}a^{12}+\frac{587198243610}{164276239249}a^{11}+\frac{65350742405883}{164276239249}a^{10}+\frac{429510882438}{164276239249}a^{9}-\frac{247904811879318}{164276239249}a^{8}-\frac{15128819844534}{164276239249}a^{7}+\frac{355219059764492}{164276239249}a^{6}+\frac{3948122573901}{164276239249}a^{5}-\frac{152623547788206}{164276239249}a^{4}+\frac{8548053366897}{164276239249}a^{3}+\frac{19632101270067}{164276239249}a^{2}-\frac{1621373189820}{164276239249}a-\frac{212054413779}{164276239249}$, $\frac{264234824073}{164276239249}a^{14}-\frac{61191766647}{164276239249}a^{13}-\frac{7079069663936}{164276239249}a^{12}+\frac{587198243610}{164276239249}a^{11}+\frac{65350742405883}{164276239249}a^{10}+\frac{429510882438}{164276239249}a^{9}-\frac{247904811879318}{164276239249}a^{8}-\frac{15128819844534}{164276239249}a^{7}+\frac{355219059764492}{164276239249}a^{6}+\frac{3948122573901}{164276239249}a^{5}-\frac{152623547788206}{164276239249}a^{4}+\frac{8548053366897}{164276239249}a^{3}+\frac{19632101270067}{164276239249}a^{2}-\frac{1621373189820}{164276239249}a-\frac{376330653028}{164276239249}$, $\frac{154206270031}{164276239249}a^{14}-\frac{128408041174}{164276239249}a^{13}-\frac{4080975970923}{164276239249}a^{12}+\frac{2820732419358}{164276239249}a^{11}+\frac{37161647641349}{164276239249}a^{10}-\frac{22648590550179}{164276239249}a^{9}-\frac{137759544898184}{164276239249}a^{8}+\frac{78559654711091}{164276239249}a^{7}+\frac{186163953295627}{164276239249}a^{6}-\frac{125512486783155}{164276239249}a^{5}-\frac{53780444299022}{164276239249}a^{4}+\frac{61330729665941}{164276239249}a^{3}-\frac{5898588981172}{164276239249}a^{2}-\frac{7937197309813}{164276239249}a+\frac{1957493944456}{164276239249}$, $\frac{640565477101}{164276239249}a^{14}-\frac{325426590720}{164276239249}a^{13}-\frac{17178805529045}{164276239249}a^{12}+\frac{6160945295434}{164276239249}a^{11}+\frac{159598868725329}{164276239249}a^{10}-\frac{42341392341765}{164276239249}a^{9}-\frac{615633040117084}{164276239249}a^{8}+\frac{124392763962720}{164276239249}a^{7}+\frac{924682931519270}{164276239249}a^{6}-\frac{206759966427839}{164276239249}a^{5}-\frac{444464619928527}{164276239249}a^{4}+\frac{97948051446399}{164276239249}a^{3}+\frac{67533645857370}{164276239249}a^{2}-\frac{11092546828131}{164276239249}a-\frac{1977657101211}{164276239249}$, $\frac{330536140}{1507121461}a^{14}+\frac{451104385}{1507121461}a^{13}-\frac{8939800589}{1507121461}a^{12}-\frac{13350270113}{1507121461}a^{11}+\frac{81896001410}{1507121461}a^{10}+\frac{129508083098}{1507121461}a^{9}-\frac{299777306206}{1507121461}a^{8}-\frac{499051484244}{1507121461}a^{7}+\frac{378760840155}{1507121461}a^{6}+\frac{656487237304}{1507121461}a^{5}-\frac{135707785479}{1507121461}a^{4}-\frac{218623066585}{1507121461}a^{3}+\frac{23311685790}{1507121461}a^{2}+\frac{16005853731}{1507121461}a-\frac{3682799320}{1507121461}$, $\frac{116151399073}{164276239249}a^{14}-\frac{197697840174}{164276239249}a^{13}-\frac{3130011144669}{164276239249}a^{12}+\frac{4833570237772}{164276239249}a^{11}+\frac{29892757322751}{164276239249}a^{10}-\frac{41811920368299}{164276239249}a^{9}-\frac{123422981848519}{164276239249}a^{8}+\frac{150098668364052}{164276239249}a^{7}+\frac{218694713053788}{164276239249}a^{6}-\frac{211873514832101}{164276239249}a^{5}-\frac{140976812824731}{164276239249}a^{4}+\frac{84325590124556}{164276239249}a^{3}+\frac{28465396049634}{164276239249}a^{2}-\frac{8791761088632}{164276239249}a-\frac{1226751444107}{164276239249}$, $\frac{292270098161}{164276239249}a^{14}-\frac{84143745533}{164276239249}a^{13}-\frac{7848355796861}{164276239249}a^{12}+\frac{1090274628733}{164276239249}a^{11}+\frac{72833705158046}{164276239249}a^{10}-\frac{3502624298576}{164276239249}a^{9}-\frac{279553281276906}{164276239249}a^{8}-\frac{2617175424007}{164276239249}a^{7}+\frac{413214035053101}{164276239249}a^{6}-\frac{11845509573861}{164276239249}a^{5}-\frac{193467073348398}{164276239249}a^{4}+\frac{12834918136824}{164276239249}a^{3}+\frac{27392540203071}{164276239249}a^{2}-\frac{1852481341690}{164276239249}a-\frac{489932787919}{164276239249}$, $\frac{39992328004}{164276239249}a^{14}+\frac{54853901655}{164276239249}a^{13}-\frac{1087659883976}{164276239249}a^{12}-\frac{1629702449832}{164276239249}a^{11}+\frac{10074930052770}{164276239249}a^{10}+\frac{15944319991099}{164276239249}a^{9}-\frac{37856101859186}{164276239249}a^{8}-\frac{62603110080682}{164276239249}a^{7}+\frac{52063164034107}{164276239249}a^{6}+\frac{87087442140829}{164276239249}a^{5}-\frac{25566063040429}{164276239249}a^{4}-\frac{35516067827051}{164276239249}a^{3}+\frac{7063639244037}{164276239249}a^{2}+\frac{3956589533414}{164276239249}a-\frac{801136868929}{164276239249}$, $\frac{247613191937}{164276239249}a^{14}-\frac{59735034653}{164276239249}a^{13}-\frac{6646675879732}{164276239249}a^{12}+\frac{615354397079}{164276239249}a^{11}+\frac{61596028556955}{164276239249}a^{10}-\frac{163569478331}{164276239249}a^{9}-\frac{235652882213627}{164276239249}a^{8}-\frac{12446424896409}{164276239249}a^{7}+\frac{345650006330807}{164276239249}a^{6}+\frac{2968131764321}{164276239249}a^{5}-\frac{160985336371908}{164276239249}a^{4}+\frac{7118682524408}{164276239249}a^{3}+\frac{25947453322985}{164276239249}a^{2}-\frac{1260821457255}{164276239249}a-\frac{1094985144409}{164276239249}$, $\frac{565169032303}{164276239249}a^{14}+\frac{42798513928}{164276239249}a^{13}-\frac{15160755188855}{164276239249}a^{12}-\frac{3411844224292}{164276239249}a^{11}+\frac{139603599912649}{164276239249}a^{10}+\frac{44181892072025}{164276239249}a^{9}-\frac{524695717822665}{164276239249}a^{8}-\frac{197599344958266}{164276239249}a^{7}+\frac{729012621875702}{164276239249}a^{6}+\frac{248713397764864}{164276239249}a^{5}-\frac{290567008686901}{164276239249}a^{4}-\frac{89783791011393}{164276239249}a^{3}+\frac{29568939555247}{164276239249}a^{2}+\frac{9640706543019}{164276239249}a+\frac{687665361962}{164276239249}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 967645.576239 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{15}\cdot(2\pi)^{0}\cdot 967645.576239 \cdot 1}{2\sqrt{10943023107606534329121}}\approx 0.151554067476$ (assuming GRH)

Galois group

$C_{15}$ (as 15T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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.

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.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.15.20.65$x^{15} + 78 x^{14} + 39 x^{13} + 49 x^{12} + 24 x^{11} + 36 x^{10} + 2 x^{9} + 54 x^{8} + 69 x^{7} + 47 x^{6} + 18 x^{5} + 15 x^{4} + 36 x^{3} + 36 x^{2} + 63 x + 73$$3$$5$$20$$C_{15}$$[2]^{5}$
\(11\) Copy content Toggle raw display 11.15.12.1$x^{15} + 165 x^{10} + 5324 x^{5} + 323433$$5$$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 *.