# 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)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining 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]);

$$x^{15} - 27 x^{13} - 4 x^{12} + 252 x^{11} + 60 x^{10} - 976 x^{9} - 288 x^{8} + 1473 x^{7} + 384 x^{6} - 765 x^{5} - 168 x^{4} + 150 x^{3} + 27 x^{2} - 9 x - 1$$

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: $$10943023107606534329121$$ 10943023107606534329121 $$\medspace = 3^{20}\cdot 11^{12}$$ 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: $$3$$, $$11$$ 3, 11 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 $$a$$)

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

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

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
$$3$$ 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$$ 11.15.12.1x^{15} + 165 x^{10} + 5324 x^{5} + 323433$$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 *.