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Properties

Label 28.28.2855088710...8125.1
Degree $28$
Signature $[28, 0]$
Discriminant $3^{14}\cdot 5^{21}\cdot 29^{24}$
Root discriminant $103.82$
Ramified primes $3, 5, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{28}$ (as 28T1)

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magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-59, 86, 57190, 199864, -2759285, -652569, 33120187, -42555117, -59689322, 114771575, 39192998, -130526538, -3555519, 82829451, -9514433, -32326986, 6204946, 8047861, -1916913, -1282732, 343545, 128066, -36878, -7603, 2304, 240, -76, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 3*x^27 - 76*x^26 + 240*x^25 + 2304*x^24 - 7603*x^23 - 36878*x^22 + 128066*x^21 + 343545*x^20 - 1282732*x^19 - 1916913*x^18 + 8047861*x^17 + 6204946*x^16 - 32326986*x^15 - 9514433*x^14 + 82829451*x^13 - 3555519*x^12 - 130526538*x^11 + 39192998*x^10 + 114771575*x^9 - 59689322*x^8 - 42555117*x^7 + 33120187*x^6 - 652569*x^5 - 2759285*x^4 + 199864*x^3 + 57190*x^2 + 86*x - 59)
 
gp: K = bnfinit(x^28 - 3*x^27 - 76*x^26 + 240*x^25 + 2304*x^24 - 7603*x^23 - 36878*x^22 + 128066*x^21 + 343545*x^20 - 1282732*x^19 - 1916913*x^18 + 8047861*x^17 + 6204946*x^16 - 32326986*x^15 - 9514433*x^14 + 82829451*x^13 - 3555519*x^12 - 130526538*x^11 + 39192998*x^10 + 114771575*x^9 - 59689322*x^8 - 42555117*x^7 + 33120187*x^6 - 652569*x^5 - 2759285*x^4 + 199864*x^3 + 57190*x^2 + 86*x - 59, 1)
 

Normalized defining polynomial

\( x^{28} - 3 x^{27} - 76 x^{26} + 240 x^{25} + 2304 x^{24} - 7603 x^{23} - 36878 x^{22} + 128066 x^{21} + 343545 x^{20} - 1282732 x^{19} - 1916913 x^{18} + 8047861 x^{17} + 6204946 x^{16} - 32326986 x^{15} - 9514433 x^{14} + 82829451 x^{13} - 3555519 x^{12} - 130526538 x^{11} + 39192998 x^{10} + 114771575 x^{9} - 59689322 x^{8} - 42555117 x^{7} + 33120187 x^{6} - 652569 x^{5} - 2759285 x^{4} + 199864 x^{3} + 57190 x^{2} + 86 x - 59 \)

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:  \(285508871014090994161304854408230095373577594757080078125=3^{14}\cdot 5^{21}\cdot 29^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $103.82$
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$$\chi_{435}(256,·)$, $\chi_{435}(1,·)$, $\chi_{435}(107,·)$, $\chi_{435}(197,·)$, $\chi_{435}(257,·)$, $\chi_{435}(136,·)$, $\chi_{435}(139,·)$, $\chi_{435}(94,·)$, $\chi_{435}(16,·)$, $\chi_{435}(248,·)$, $\chi_{435}(83,·)$, $\chi_{435}(407,·)$, $\chi_{435}(152,·)$, $\chi_{435}(413,·)$, $\chi_{435}(286,·)$, $\chi_{435}(226,·)$, $\chi_{435}(227,·)$, $\chi_{435}(422,·)$, $\chi_{435}(169,·)$, $\chi_{435}(199,·)$, $\chi_{435}(364,·)$, $\chi_{435}(349,·)$, $\chi_{435}(368,·)$, $\chi_{435}(49,·)$, $\chi_{435}(53,·)$, $\chi_{435}(233,·)$, $\chi_{435}(23,·)$, $\chi_{435}(181,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{1003} a^{24} + \frac{309}{1003} a^{23} - \frac{269}{1003} a^{22} - \frac{11}{1003} a^{21} + \frac{439}{1003} a^{20} - \frac{12}{59} a^{19} - \frac{1}{1003} a^{18} - \frac{429}{1003} a^{17} - \frac{11}{59} a^{16} + \frac{330}{1003} a^{15} - \frac{322}{1003} a^{14} + \frac{91}{1003} a^{13} - \frac{350}{1003} a^{12} + \frac{55}{1003} a^{11} + \frac{130}{1003} a^{10} + \frac{184}{1003} a^{9} + \frac{104}{1003} a^{8} - \frac{331}{1003} a^{7} + \frac{77}{1003} a^{6} + \frac{100}{1003} a^{5} + \frac{477}{1003} a^{4} - \frac{216}{1003} a^{3} + \frac{432}{1003} a^{2} + \frac{358}{1003} a + \frac{6}{17}$, $\frac{1}{1003} a^{25} - \frac{465}{1003} a^{23} - \frac{139}{1003} a^{22} - \frac{174}{1003} a^{21} - \frac{450}{1003} a^{20} - \frac{154}{1003} a^{19} - \frac{120}{1003} a^{18} - \frac{22}{1003} a^{17} - \frac{61}{1003} a^{16} + \frac{14}{1003} a^{15} + \frac{292}{1003} a^{14} - \frac{385}{1003} a^{13} - \frac{7}{59} a^{12} + \frac{186}{1003} a^{11} + \frac{134}{1003} a^{10} + \frac{419}{1003} a^{9} - \frac{371}{1003} a^{8} + \frac{50}{1003} a^{7} + \frac{379}{1003} a^{6} - \frac{333}{1003} a^{5} - \frac{168}{1003} a^{4} - \frac{25}{1003} a^{3} + \frac{269}{1003} a^{2} + \frac{62}{1003} a - \frac{1}{17}$, $\frac{1}{1003} a^{26} + \frac{117}{1003} a^{23} + \frac{116}{1003} a^{22} + \frac{453}{1003} a^{21} + \frac{372}{1003} a^{20} + \frac{305}{1003} a^{19} - \frac{487}{1003} a^{18} + \frac{3}{59} a^{17} + \frac{320}{1003} a^{16} + \frac{283}{1003} a^{15} + \frac{335}{1003} a^{14} + \frac{70}{1003} a^{13} - \frac{78}{1003} a^{12} - \frac{369}{1003} a^{11} - \frac{314}{1003} a^{10} - \frac{66}{1003} a^{9} + \frac{266}{1003} a^{8} - \frac{77}{1003} a^{7} + \frac{367}{1003} a^{6} + \frac{194}{1003} a^{5} + \frac{117}{1003} a^{4} + \frac{129}{1003} a^{3} + \frac{342}{1003} a^{2} - \frac{87}{1003} a + \frac{2}{17}$, $\frac{1}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{27} + \frac{2346979375000998260322794874628667187557338332893455294514034632249113}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{26} + \frac{2142156255268386425731493822245901033313534339157932269894259657228815}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{25} + \frac{926485047979097795992964306750191813863526072345959006810049209698108}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{24} + \frac{4074371268286333007055044596078112636235334888886114795141683968867290294}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{23} - \frac{3038268121864730533738301345818214170507255819844521902154241924315527}{24795033662144760707296675031155014417194287111261091578360919332672543} a^{22} + \frac{2942677109492149458567714378947424373236152840253159381904458612333801948}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{21} - \frac{1239113470620506572118111450742601993421360695803035545776349886808047853}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{20} + \frac{1987980026940462953711683345750289670985162010001741826363565151038722568}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{19} + \frac{3982425366130634386827026778789449139400700985769919752865056150207231088}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{18} + \frac{1542776960777688181583199754540432508816922534834195090097140368884281659}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{17} - \frac{2007903081277259296230820486493007410843949060686612595713943610555974277}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{16} - \frac{3675994508844762180511220825096291650400579860148332165429933422600841793}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{15} + \frac{184420999772698154273025133267523812560834906628653975793889031068339190}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{14} - \frac{3965265711605178392419356996371791854599113713826665950669394022333933622}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{13} - \frac{2960817660309543146382206772347077752555479849834338082425389583667954738}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{12} - \frac{3161702126518385229878656723238130418773856337471110266157477256691556894}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{11} + \frac{3678697023337933990411081325967943016383860817927161200523348410799059506}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{10} + \frac{3460138039548557967384276852450570763331150556703906474448940768010689978}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{9} - \frac{2734840523860435206891522402485893463243718916433273863924340128953052419}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{8} - \frac{2920004423358987176659452946378202843420654948009087217151856976992867841}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{7} + \frac{685389177484688280979924887854453672216857154119781628646669649177471745}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{6} + \frac{3553056926487609525002557230540311731198350647205435878427640922529947509}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{5} - \frac{2652461999645259590413981548434169063829145576222311716143640624993406262}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{4} - \frac{720981617306118826888625832762176919808559163125149455312618940384718024}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{3} + \frac{2659383582071348228511855541255202798715036981245336657558050526362718287}{8207156142169915794115199435312309772091309033827421312437464299114611733} a^{2} - \frac{3630297823651341724498454702278597435730185850167926933069626823154632453}{8207156142169915794115199435312309772091309033827421312437464299114611733} a + \frac{32293147062636812693961750353803750306473673080072912707863814877977868}{139104341392710437188393210768005250374428966675041039193855327103637487}$

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:  \( 23509806182109426000 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 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 $28$ R R $28$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ $28$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ $28$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ $28$ $28$ $28$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

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
3Data not computed
5Data not computed
$29$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$