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Properties

Label 28.0.30734458946...5625.3
Degree $28$
Signature $[0, 14]$
Discriminant $3^{14}\cdot 5^{14}\cdot 29^{26}$
Root discriminant $88.31$
Ramified primes $3, 5, 29$
Class number $173568$ (GRH)
Class group $[4, 4, 4, 2712]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![297525211, 1702686, 585050360, 346864593, 281593991, 538783144, 215771335, 131564944, 234860782, 8840615, 99880149, 1747694, 43894811, -1380551, 13304046, -1269653, 2881062, -293864, 517055, -39323, 73123, -3706, 8185, -337, 682, -22, 37, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 37*x^26 - 22*x^25 + 682*x^24 - 337*x^23 + 8185*x^22 - 3706*x^21 + 73123*x^20 - 39323*x^19 + 517055*x^18 - 293864*x^17 + 2881062*x^16 - 1269653*x^15 + 13304046*x^14 - 1380551*x^13 + 43894811*x^12 + 1747694*x^11 + 99880149*x^10 + 8840615*x^9 + 234860782*x^8 + 131564944*x^7 + 215771335*x^6 + 538783144*x^5 + 281593991*x^4 + 346864593*x^3 + 585050360*x^2 + 1702686*x + 297525211)
 
gp: K = bnfinit(x^28 - x^27 + 37*x^26 - 22*x^25 + 682*x^24 - 337*x^23 + 8185*x^22 - 3706*x^21 + 73123*x^20 - 39323*x^19 + 517055*x^18 - 293864*x^17 + 2881062*x^16 - 1269653*x^15 + 13304046*x^14 - 1380551*x^13 + 43894811*x^12 + 1747694*x^11 + 99880149*x^10 + 8840615*x^9 + 234860782*x^8 + 131564944*x^7 + 215771335*x^6 + 538783144*x^5 + 281593991*x^4 + 346864593*x^3 + 585050360*x^2 + 1702686*x + 297525211, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} + 37 x^{26} - 22 x^{25} + 682 x^{24} - 337 x^{23} + 8185 x^{22} - 3706 x^{21} + 73123 x^{20} - 39323 x^{19} + 517055 x^{18} - 293864 x^{17} + 2881062 x^{16} - 1269653 x^{15} + 13304046 x^{14} - 1380551 x^{13} + 43894811 x^{12} + 1747694 x^{11} + 99880149 x^{10} + 8840615 x^{9} + 234860782 x^{8} + 131564944 x^{7} + 215771335 x^{6} + 538783144 x^{5} + 281593991 x^{4} + 346864593 x^{3} + 585050360 x^{2} + 1702686 x + 297525211 \)

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

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 14]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3073445894692486733947614496733715330677488092041015625=3^{14}\cdot 5^{14}\cdot 29^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.31$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(435=3\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{435}(256,·)$, $\chi_{435}(1,·)$, $\chi_{435}(386,·)$, $\chi_{435}(4,·)$, $\chi_{435}(71,·)$, $\chi_{435}(136,·)$, $\chi_{435}(226,·)$, $\chi_{435}(266,·)$, $\chi_{435}(194,·)$, $\chi_{435}(16,·)$, $\chi_{435}(274,·)$, $\chi_{435}(341,·)$, $\chi_{435}(86,·)$, $\chi_{435}(344,·)$, $\chi_{435}(154,·)$, $\chi_{435}(284,·)$, $\chi_{435}(286,·)$, $\chi_{435}(289,·)$, $\chi_{435}(34,·)$, $\chi_{435}(296,·)$, $\chi_{435}(64,·)$, $\chi_{435}(236,·)$, $\chi_{435}(109,·)$, $\chi_{435}(239,·)$, $\chi_{435}(181,·)$, $\chi_{435}(314,·)$, $\chi_{435}(59,·)$, $\chi_{435}(74,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{17} a^{16} - \frac{2}{17} a^{15} + \frac{4}{17} a^{14} - \frac{8}{17} a^{13} - \frac{1}{17} a^{12} + \frac{2}{17} a^{11} - \frac{4}{17} a^{10} + \frac{8}{17} a^{9} + \frac{1}{17} a^{8} - \frac{2}{17} a^{7} + \frac{4}{17} a^{6} - \frac{8}{17} a^{5} - \frac{1}{17} a^{4} + \frac{2}{17} a^{3} - \frac{4}{17} a^{2} + \frac{8}{17} a$, $\frac{1}{17} a^{17} - \frac{1}{17} a$, $\frac{1}{17} a^{18} - \frac{1}{17} a^{2}$, $\frac{1}{17} a^{19} - \frac{1}{17} a^{3}$, $\frac{1}{17} a^{20} - \frac{1}{17} a^{4}$, $\frac{1}{34} a^{21} - \frac{1}{34} a^{19} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{34} a^{5} - \frac{8}{17} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{34} a^{22} - \frac{1}{34} a^{20} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{34} a^{6} - \frac{8}{17} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{34} a^{23} - \frac{1}{34} a^{19} - \frac{1}{34} a^{16} - \frac{15}{34} a^{15} - \frac{2}{17} a^{14} + \frac{4}{17} a^{13} + \frac{1}{34} a^{12} - \frac{1}{17} a^{11} + \frac{2}{17} a^{10} + \frac{9}{34} a^{9} + \frac{8}{17} a^{8} + \frac{1}{34} a^{7} - \frac{2}{17} a^{6} - \frac{9}{34} a^{5} + \frac{1}{34} a^{4} - \frac{1}{34} a^{3} + \frac{2}{17} a^{2} + \frac{9}{34} a - \frac{1}{2}$, $\frac{1}{578} a^{24} + \frac{1}{578} a^{23} + \frac{5}{578} a^{22} + \frac{5}{578} a^{21} - \frac{6}{289} a^{20} + \frac{3}{289} a^{19} + \frac{1}{289} a^{18} + \frac{3}{578} a^{17} - \frac{3}{289} a^{16} + \frac{6}{289} a^{15} + \frac{141}{289} a^{14} - \frac{71}{578} a^{13} + \frac{23}{578} a^{12} - \frac{29}{578} a^{11} - \frac{124}{289} a^{10} + \frac{27}{289} a^{9} - \frac{109}{578} a^{8} + \frac{11}{578} a^{7} - \frac{23}{289} a^{6} + \frac{145}{578} a^{5} - \frac{76}{289} a^{4} - \frac{43}{289} a^{3} - \frac{63}{578} a^{2} + \frac{7}{17} a$, $\frac{1}{578} a^{25} + \frac{2}{289} a^{23} - \frac{8}{289} a^{20} + \frac{13}{578} a^{19} + \frac{1}{578} a^{18} - \frac{9}{578} a^{17} - \frac{8}{289} a^{16} - \frac{120}{289} a^{15} - \frac{100}{289} a^{14} + \frac{77}{578} a^{13} + \frac{271}{578} a^{12} + \frac{1}{289} a^{11} - \frac{70}{289} a^{10} - \frac{73}{289} a^{9} + \frac{43}{289} a^{8} + \frac{11}{578} a^{7} + \frac{55}{578} a^{6} - \frac{21}{289} a^{5} + \frac{67}{289} a^{4} + \frac{227}{578} a^{3} + \frac{74}{289} a^{2} - \frac{13}{34} a - \frac{1}{2}$, $\frac{1}{3286016579198} a^{26} - \frac{514787722}{1643008289599} a^{25} + \frac{533533213}{1643008289599} a^{24} + \frac{20063844897}{3286016579198} a^{23} - \frac{28219275}{2075815906} a^{22} + \frac{22637597684}{1643008289599} a^{21} + \frac{29126159820}{1643008289599} a^{20} + \frac{36047658067}{1643008289599} a^{19} + \frac{50308204383}{3286016579198} a^{18} - \frac{9788839743}{1643008289599} a^{17} + \frac{782195859}{3286016579198} a^{16} - \frac{289184328520}{1643008289599} a^{15} - \frac{518680696770}{1643008289599} a^{14} + \frac{183806472336}{1643008289599} a^{13} + \frac{687559457251}{1643008289599} a^{12} - \frac{541404256653}{1643008289599} a^{11} + \frac{265417025153}{3286016579198} a^{10} - \frac{1272579654217}{3286016579198} a^{9} + \frac{1585674463579}{3286016579198} a^{8} + \frac{539165243081}{1643008289599} a^{7} + \frac{1019246989899}{3286016579198} a^{6} + \frac{84497852091}{3286016579198} a^{5} + \frac{282082217695}{1643008289599} a^{4} - \frac{808500973335}{1643008289599} a^{3} - \frac{392355894157}{1643008289599} a^{2} - \frac{8877957427}{193295092894} a + \frac{2013509361}{11370299582}$, $\frac{1}{49265961483608462242005939075962305073336802232350413742730138620859137361201060680158879092191881439813542} a^{27} - \frac{3234543429724426242259289806940785138219418386331839781844136389904591096564550879562427176025}{49265961483608462242005939075962305073336802232350413742730138620859137361201060680158879092191881439813542} a^{26} - \frac{11324173107024861083444769028128217573621210349770278656332205389733017004867743239436702160993855492051}{24632980741804231121002969537981152536668401116175206871365069310429568680600530340079439546095940719906771} a^{25} + \frac{25689513997492063739076225587651530164042596332098293332528184368131586904215063528483643445567846184245}{49265961483608462242005939075962305073336802232350413742730138620859137361201060680158879092191881439813542} a^{24} + \frac{267097376415940357658757738457555164284573329245019221609925613513526866711408235165062459348036803457393}{24632980741804231121002969537981152536668401116175206871365069310429568680600530340079439546095940719906771} a^{23} - \frac{191768616779355809384639306578257759303079894367132215781380642840648472528392413420651992628492149795763}{49265961483608462242005939075962305073336802232350413742730138620859137361201060680158879092191881439813542} a^{22} + \frac{28992598602915196487152615891037372896597677321485240307105493216804171346758483864564974563550785069341}{24632980741804231121002969537981152536668401116175206871365069310429568680600530340079439546095940719906771} a^{21} + \frac{471698277284280193588618922101431486430444783169238050173781202452322344455416193274227305938561621984366}{24632980741804231121002969537981152536668401116175206871365069310429568680600530340079439546095940719906771} a^{20} + \frac{1174519792060901381114494885763005921543091843434609502022180047072467282056248567893167971652488487292167}{49265961483608462242005939075962305073336802232350413742730138620859137361201060680158879092191881439813542} a^{19} + \frac{733151996849844083685834076726402013430764657456925291674526766395232756711013795853830565458384359387717}{49265961483608462242005939075962305073336802232350413742730138620859137361201060680158879092191881439813542} a^{18} - \frac{310234772039423601316939252740802348409041001193298288977978261425808246499226124054336329398509873853811}{49265961483608462242005939075962305073336802232350413742730138620859137361201060680158879092191881439813542} a^{17} - \frac{851895271342967426901914597360901906496199878395069742177567316823973470161009425822341594056138335472995}{49265961483608462242005939075962305073336802232350413742730138620859137361201060680158879092191881439813542} a^{16} - \frac{1436830820456779078901705211524000081426532641540433280348460333310295297873253457976271128011567788450602}{24632980741804231121002969537981152536668401116175206871365069310429568680600530340079439546095940719906771} a^{15} + \frac{427429889389474619287822674832604847479074550803021643930555505048470677387468044667029590258790968774147}{24632980741804231121002969537981152536668401116175206871365069310429568680600530340079439546095940719906771} a^{14} + \frac{9699962340933076582827944982815521609128904468867832707668730918852238500074205235103773888392972651612619}{24632980741804231121002969537981152536668401116175206871365069310429568680600530340079439546095940719906771} a^{13} + \frac{518162484018760969969728443652474357713117599968788510034893135120659924994663807867309208427855381073184}{24632980741804231121002969537981152536668401116175206871365069310429568680600530340079439546095940719906771} a^{12} + \frac{21257261976448300833825933549620212772757556018393748772532237147628890804622337085348769075071156260820129}{49265961483608462242005939075962305073336802232350413742730138620859137361201060680158879092191881439813542} a^{11} + \frac{9701606001227711752500090400417751508030267802284478006707941643532727938270529875383930733098330037222575}{24632980741804231121002969537981152536668401116175206871365069310429568680600530340079439546095940719906771} a^{10} + \frac{12107626562326094550885632815181942504901584356202709263340268670151660433056135086950399926897112503057626}{24632980741804231121002969537981152536668401116175206871365069310429568680600530340079439546095940719906771} a^{9} - \frac{126454907501287984196624887009334767074754139010101397055352304213449428597256421681405161792701314798371}{313795933016614409184751204305492389002145237148728749953695150451332085103191469300375026064916442291806} a^{8} + \frac{8811397944639902985639651869358903077993795609254958492039883165190487857653930280861925226119098216688121}{49265961483608462242005939075962305073336802232350413742730138620859137361201060680158879092191881439813542} a^{7} - \frac{8160572022055867262929835557456915329306763824521104355450460699350340355679320511050555424125595965168827}{24632980741804231121002969537981152536668401116175206871365069310429568680600530340079439546095940719906771} a^{6} - \frac{4866445787708714903836646400432014720936478042596141254534603854401710987329331756573392166077077262205551}{49265961483608462242005939075962305073336802232350413742730138620859137361201060680158879092191881439813542} a^{5} - \frac{6498372262071145536580847591880817939423361619410952431402326692845688314579600641901059755445473941558339}{24632980741804231121002969537981152536668401116175206871365069310429568680600530340079439546095940719906771} a^{4} - \frac{7271628890746395325511679230856216280069927196980128704378953031895438540263084589541165618460931997600079}{24632980741804231121002969537981152536668401116175206871365069310429568680600530340079439546095940719906771} a^{3} + \frac{5091781359846008735548019982338867480231021453569345756204796558530338681496021600328419845624208147282051}{49265961483608462242005939075962305073336802232350413742730138620859137361201060680158879092191881439813542} a^{2} + \frac{67880262276356622466550468976138005325528714484254067490897476811169223366923126880363549865067613099926}{1448998867164954771823704090469479560980494183304423933609709959437033451800031196475261149770349454112163} a - \frac{821294505592811378942734699436706060471750738090139013295246965408535282330850024509670064772381690615}{170470454960582914332200481231703477762411080388755756895259995227886288447062493702971899972982288719078}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{4}\times C_{2712}$, which has order $173568$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 73869644668.60387 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{14}$ (as 28T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{-87}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-15}, \sqrt{-87})\), 7.7.594823321.1, 14.14.801611618199890796015625.1, 14.0.22439994995240462987343.1, 14.0.60452572724246936927109375.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{4}$ R R ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{28}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
5Data not computed
29Data not computed