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Properties

Label 28.28.4048504262...3125.1
Degree $28$
Signature $[28, 0]$
Discriminant $5^{21}\cdot 7^{14}\cdot 29^{24}$
Root discriminant $158.58$
Ramified primes $5, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{28}$ (as 28T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9342001, 137225481, 59014540, -1069073901, -463454290, 3489342686, 902850052, -6078001902, -603585672, 6204587325, -137358612, -3886142713, 426084076, 1531368941, -261479693, -386151006, 82351466, 62805921, -15281943, -6558612, 1739875, 430221, -121463, -16873, 5014, 355, -111, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 3*x^27 - 111*x^26 + 355*x^25 + 5014*x^24 - 16873*x^23 - 121463*x^22 + 430221*x^21 + 1739875*x^20 - 6558612*x^19 - 15281943*x^18 + 62805921*x^17 + 82351466*x^16 - 386151006*x^15 - 261479693*x^14 + 1531368941*x^13 + 426084076*x^12 - 3886142713*x^11 - 137358612*x^10 + 6204587325*x^9 - 603585672*x^8 - 6078001902*x^7 + 902850052*x^6 + 3489342686*x^5 - 463454290*x^4 - 1069073901*x^3 + 59014540*x^2 + 137225481*x + 9342001)
 
gp: K = bnfinit(x^28 - 3*x^27 - 111*x^26 + 355*x^25 + 5014*x^24 - 16873*x^23 - 121463*x^22 + 430221*x^21 + 1739875*x^20 - 6558612*x^19 - 15281943*x^18 + 62805921*x^17 + 82351466*x^16 - 386151006*x^15 - 261479693*x^14 + 1531368941*x^13 + 426084076*x^12 - 3886142713*x^11 - 137358612*x^10 + 6204587325*x^9 - 603585672*x^8 - 6078001902*x^7 + 902850052*x^6 + 3489342686*x^5 - 463454290*x^4 - 1069073901*x^3 + 59014540*x^2 + 137225481*x + 9342001, 1)
 

Normalized defining polynomial

\( x^{28} - 3 x^{27} - 111 x^{26} + 355 x^{25} + 5014 x^{24} - 16873 x^{23} - 121463 x^{22} + 430221 x^{21} + 1739875 x^{20} - 6558612 x^{19} - 15281943 x^{18} + 62805921 x^{17} + 82351466 x^{16} - 386151006 x^{15} - 261479693 x^{14} + 1531368941 x^{13} + 426084076 x^{12} - 3886142713 x^{11} - 137358612 x^{10} + 6204587325 x^{9} - 603585672 x^{8} - 6078001902 x^{7} + 902850052 x^{6} + 3489342686 x^{5} - 463454290 x^{4} - 1069073901 x^{3} + 59014540 x^{2} + 137225481 x + 9342001 \)

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

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[28, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(40485042622025269417087481839880287486644411644458770751953125=5^{21}\cdot 7^{14}\cdot 29^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $158.58$
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:  $5, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1015=5\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1015}(832,·)$, $\chi_{1015}(1,·)$, $\chi_{1015}(484,·)$, $\chi_{1015}(132,·)$, $\chi_{1015}(517,·)$, $\chi_{1015}(587,·)$, $\chi_{1015}(204,·)$, $\chi_{1015}(141,·)$, $\chi_{1015}(83,·)$, $\chi_{1015}(596,·)$, $\chi_{1015}(981,·)$, $\chi_{1015}(342,·)$, $\chi_{1015}(344,·)$, $\chi_{1015}(281,·)$, $\chi_{1015}(538,·)$, $\chi_{1015}(923,·)$, $\chi_{1015}(799,·)$, $\chi_{1015}(993,·)$, $\chi_{1015}(36,·)$, $\chi_{1015}(806,·)$, $\chi_{1015}(552,·)$, $\chi_{1015}(169,·)$, $\chi_{1015}(748,·)$, $\chi_{1015}(239,·)$, $\chi_{1015}(1009,·)$, $\chi_{1015}(692,·)$, $\chi_{1015}(223,·)$, $\chi_{1015}(958,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{59} a^{23} - \frac{25}{59} a^{22} - \frac{1}{59} a^{21} - \frac{4}{59} a^{20} + \frac{21}{59} a^{19} + \frac{29}{59} a^{18} + \frac{17}{59} a^{17} - \frac{11}{59} a^{16} + \frac{3}{59} a^{15} + \frac{3}{59} a^{14} - \frac{2}{59} a^{13} - \frac{25}{59} a^{12} - \frac{1}{59} a^{11} + \frac{23}{59} a^{10} + \frac{23}{59} a^{9} - \frac{2}{59} a^{8} + \frac{22}{59} a^{7} - \frac{6}{59} a^{6} - \frac{7}{59} a^{5} + \frac{27}{59} a^{4} - \frac{18}{59} a^{3} + \frac{29}{59} a^{2} + \frac{17}{59} a$, $\frac{1}{1003} a^{24} + \frac{3}{1003} a^{23} + \frac{7}{1003} a^{22} + \frac{440}{1003} a^{21} + \frac{381}{1003} a^{20} + \frac{381}{1003} a^{19} + \frac{416}{1003} a^{18} - \frac{7}{1003} a^{17} + \frac{167}{1003} a^{16} + \frac{146}{1003} a^{15} + \frac{318}{1003} a^{14} - \frac{376}{1003} a^{13} - \frac{229}{1003} a^{12} + \frac{467}{1003} a^{11} + \frac{490}{1003} a^{10} - \frac{184}{1003} a^{9} + \frac{320}{1003} a^{8} + \frac{197}{1003} a^{7} + \frac{2}{1003} a^{6} + \frac{67}{1003} a^{5} + \frac{266}{1003} a^{4} + \frac{351}{1003} a^{3} + \frac{21}{59} a^{2} + \frac{122}{1003} a - \frac{3}{17}$, $\frac{1}{1003} a^{25} - \frac{2}{1003} a^{23} + \frac{419}{1003} a^{22} + \frac{64}{1003} a^{21} + \frac{241}{1003} a^{20} + \frac{276}{1003} a^{19} - \frac{252}{1003} a^{18} + \frac{188}{1003} a^{17} - \frac{355}{1003} a^{16} - \frac{120}{1003} a^{15} - \frac{327}{1003} a^{14} - \frac{104}{1003} a^{13} + \frac{151}{1003} a^{12} + \frac{92}{1003} a^{11} + \frac{352}{1003} a^{10} - \frac{131}{1003} a^{9} + \frac{240}{1003} a^{8} + \frac{414}{1003} a^{7} + \frac{61}{1003} a^{6} + \frac{65}{1003} a^{5} - \frac{447}{1003} a^{4} + \frac{307}{1003} a^{3} + \frac{54}{1003} a^{2} + \frac{460}{1003} a - \frac{8}{17}$, $\frac{1}{331993} a^{26} - \frac{1}{19529} a^{25} + \frac{121}{331993} a^{24} - \frac{2561}{331993} a^{23} + \frac{61326}{331993} a^{22} - \frac{28599}{331993} a^{21} - \frac{39714}{331993} a^{20} + \frac{966}{331993} a^{19} + \frac{15707}{331993} a^{18} + \frac{60443}{331993} a^{17} - \frac{154985}{331993} a^{16} - \frac{128892}{331993} a^{15} + \frac{1321}{331993} a^{14} - \frac{48596}{331993} a^{13} - \frac{23298}{331993} a^{12} + \frac{108759}{331993} a^{11} + \frac{134820}{331993} a^{10} - \frac{6701}{331993} a^{9} - \frac{4681}{331993} a^{8} + \frac{9026}{331993} a^{7} + \frac{161998}{331993} a^{6} + \frac{5295}{331993} a^{5} + \frac{128820}{331993} a^{4} - \frac{118749}{331993} a^{3} - \frac{100792}{331993} a^{2} - \frac{30737}{331993} a + \frac{1620}{5627}$, $\frac{1}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{27} + \frac{103534624477800512879542924762904535181832003086224146666316778874396376056863870140265561}{348593417786525303553133401758419468070249900948308723892776230801676988292901825076758556033801} a^{26} - \frac{1889029036090666522328325311262863911007879769257763981989924715249065681696018670495967609897}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{25} - \frac{1668158169483516201080884735736266509363010204639930930810033430266666550549315336715908580321}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{24} - \frac{5535892029145093419472394780484086126971819553914554856010005926910821640351288474401021657137}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{23} - \frac{380649023269827339726112118590558499635259750502224893074134840006697446185096550326252017820792}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{22} + \frac{2343831789705724472729635847399990207927891174033471040824941192235017123116080244776708442943087}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{21} + \frac{1314922332546218774189993663358288828275968070371209696565254766992853287082790840200679012203910}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{20} + \frac{435145673427045977894302205241918171500755508931059147426733335341950551563096295331195711416414}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{19} + \frac{570537283985891742203964524514711644356435815533007238514779502505613789750557254725666151572367}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{18} + \frac{2125987846320629222221566136209361834219762787466870707769702391011856357771075620366206999100266}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{17} + \frac{476652957466156167752672160566203524869787908722204776040085541269506773043728746252478296591008}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{16} + \frac{1442486646552121369830548770620591649866252401145292515071398972361198588383161353164997647606203}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{15} + \frac{1203790289261999954539330598938285432580705317338456227765325711131244109369370278094253756927027}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{14} + \frac{2065523747784132321980560774648625818329817241712408988429570407429559610769698424195078339336804}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{13} - \frac{1204861245224070047218751880035183909411639813109968120675673246881983091195764549315597760219134}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{12} - \frac{419475667859981496479491762217556022482418592196450130212620440551531287965388335091989009674239}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{11} + \frac{1733984851559024176307842322614658266477537166850457989869241400023867204564760254456022151118895}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{10} + \frac{1759832552872249231410200265739900114253073162354459558763041682650765198635619455365327658318310}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{9} - \frac{149327356584916328267936281568975111850415481203139060933418678504664120810397039989499267287784}{348593417786525303553133401758419468070249900948308723892776230801676988292901825076758556033801} a^{8} - \frac{1958247851933464218158965265822549516811820412273568340076234824833399242454211581950600841424160}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{7} - \frac{1598519195969463121451270845670569962190806891331664628029595522404874555187197823308919948433118}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{6} - \frac{60885897379814615723737196076400060155911989246176352750456846276958731124939628297999626536618}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{5} + \frac{1824027315960443516536935852453642806626608976327994305394034007772161914472663618387299488323720}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{4} - \frac{593191274596999085586061257415375467278718521707182843263829281606925442047278238103696721012393}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{3} - \frac{2802331527564842546628292965281379468282724830331982096140869711978497535746693816737419596388995}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a^{2} - \frac{778258572267784684625254941572115865525529526966833934536285465938084635039956794294279591245970}{5926088102370930160403267829893130957194248316121248306177195923628508800979331026304895452574617} a + \frac{3019273471969022383499822948469464990995040290211001142315723281677569799126033809041612371857}{100442171226625934922089285252425948427021157900360140782664337688618793236937814005167719535163}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $27$
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:  \( 9549725081294820000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{28}$ (as 28T1):

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

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.6125.1, 7.7.594823321.1, 14.14.27641779937927268828125.1

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 $28$ $28$ R R ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{4}$ $28$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ $28$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ $28$ $28$ $28$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

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
5Data not computed
7Data not computed
$29$29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$
29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$