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Properties

Label 28.28.8575749600...3125.1
Degree $28$
Signature $[28, 0]$
Discriminant $5^{21}\cdot 7^{50}$
Root discriminant $107.98$
Ramified primes $5, 7$
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![361, 17822, 194915, 540274, -1496096, -6990543, 2492042, 30363184, 7294518, -63232365, -31909787, 70220969, 45977568, -44286151, -33849902, 16439787, 14278698, -3640098, -3616123, 476021, 558551, -35384, -52311, 1365, 2870, -21, -84, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 84*x^26 - 21*x^25 + 2870*x^24 + 1365*x^23 - 52311*x^22 - 35384*x^21 + 558551*x^20 + 476021*x^19 - 3616123*x^18 - 3640098*x^17 + 14278698*x^16 + 16439787*x^15 - 33849902*x^14 - 44286151*x^13 + 45977568*x^12 + 70220969*x^11 - 31909787*x^10 - 63232365*x^9 + 7294518*x^8 + 30363184*x^7 + 2492042*x^6 - 6990543*x^5 - 1496096*x^4 + 540274*x^3 + 194915*x^2 + 17822*x + 361)
 
gp: K = bnfinit(x^28 - 84*x^26 - 21*x^25 + 2870*x^24 + 1365*x^23 - 52311*x^22 - 35384*x^21 + 558551*x^20 + 476021*x^19 - 3616123*x^18 - 3640098*x^17 + 14278698*x^16 + 16439787*x^15 - 33849902*x^14 - 44286151*x^13 + 45977568*x^12 + 70220969*x^11 - 31909787*x^10 - 63232365*x^9 + 7294518*x^8 + 30363184*x^7 + 2492042*x^6 - 6990543*x^5 - 1496096*x^4 + 540274*x^3 + 194915*x^2 + 17822*x + 361, 1)
 

Normalized defining polynomial

\( x^{28} - 84 x^{26} - 21 x^{25} + 2870 x^{24} + 1365 x^{23} - 52311 x^{22} - 35384 x^{21} + 558551 x^{20} + 476021 x^{19} - 3616123 x^{18} - 3640098 x^{17} + 14278698 x^{16} + 16439787 x^{15} - 33849902 x^{14} - 44286151 x^{13} + 45977568 x^{12} + 70220969 x^{11} - 31909787 x^{10} - 63232365 x^{9} + 7294518 x^{8} + 30363184 x^{7} + 2492042 x^{6} - 6990543 x^{5} - 1496096 x^{4} + 540274 x^{3} + 194915 x^{2} + 17822 x + 361 \)

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:  \(857574960063654015836849375042748140931725025177001953125=5^{21}\cdot 7^{50}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $107.98$
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:  $5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(245=5\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{245}(64,·)$, $\chi_{245}(1,·)$, $\chi_{245}(132,·)$, $\chi_{245}(134,·)$, $\chi_{245}(71,·)$, $\chi_{245}(202,·)$, $\chi_{245}(204,·)$, $\chi_{245}(13,·)$, $\chi_{245}(141,·)$, $\chi_{245}(83,·)$, $\chi_{245}(97,·)$, $\chi_{245}(153,·)$, $\chi_{245}(27,·)$, $\chi_{245}(29,·)$, $\chi_{245}(223,·)$, $\chi_{245}(48,·)$, $\chi_{245}(99,·)$, $\chi_{245}(36,·)$, $\chi_{245}(167,·)$, $\chi_{245}(169,·)$, $\chi_{245}(106,·)$, $\chi_{245}(237,·)$, $\chi_{245}(239,·)$, $\chi_{245}(176,·)$, $\chi_{245}(211,·)$, $\chi_{245}(118,·)$, $\chi_{245}(188,·)$, $\chi_{245}(62,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{19} a^{13} - \frac{8}{19} a^{12} - \frac{4}{19} a^{11} - \frac{5}{19} a^{10} - \frac{6}{19} a^{8} - \frac{3}{19} a^{7} - \frac{5}{19} a^{6} - \frac{9}{19} a^{5} - \frac{8}{19} a^{4} + \frac{6}{19} a^{3} - \frac{7}{19} a^{2} + \frac{8}{19} a$, $\frac{1}{19} a^{14} + \frac{8}{19} a^{12} + \frac{1}{19} a^{11} - \frac{2}{19} a^{10} - \frac{6}{19} a^{9} + \frac{6}{19} a^{8} + \frac{9}{19} a^{7} + \frac{8}{19} a^{6} - \frac{4}{19} a^{5} - \frac{1}{19} a^{4} + \frac{3}{19} a^{3} + \frac{9}{19} a^{2} + \frac{7}{19} a$, $\frac{1}{19} a^{15} + \frac{8}{19} a^{12} - \frac{8}{19} a^{11} - \frac{4}{19} a^{10} + \frac{6}{19} a^{9} - \frac{6}{19} a^{7} - \frac{2}{19} a^{6} - \frac{5}{19} a^{5} - \frac{9}{19} a^{4} - \frac{1}{19} a^{3} + \frac{6}{19} a^{2} - \frac{7}{19} a$, $\frac{1}{19} a^{16} - \frac{1}{19} a^{12} + \frac{9}{19} a^{11} + \frac{8}{19} a^{10} + \frac{4}{19} a^{8} + \frac{3}{19} a^{7} - \frac{3}{19} a^{6} + \frac{6}{19} a^{5} + \frac{6}{19} a^{4} - \frac{4}{19} a^{3} - \frac{8}{19} a^{2} - \frac{7}{19} a$, $\frac{1}{19} a^{17} + \frac{1}{19} a^{12} + \frac{4}{19} a^{11} - \frac{5}{19} a^{10} + \frac{4}{19} a^{9} - \frac{3}{19} a^{8} - \frac{6}{19} a^{7} + \frac{1}{19} a^{6} - \frac{3}{19} a^{5} + \frac{7}{19} a^{4} - \frac{2}{19} a^{3} + \frac{5}{19} a^{2} + \frac{8}{19} a$, $\frac{1}{19} a^{18} - \frac{7}{19} a^{12} - \frac{1}{19} a^{11} + \frac{9}{19} a^{10} - \frac{3}{19} a^{9} + \frac{4}{19} a^{7} + \frac{2}{19} a^{6} - \frac{3}{19} a^{5} + \frac{6}{19} a^{4} - \frac{1}{19} a^{3} - \frac{4}{19} a^{2} - \frac{8}{19} a$, $\frac{1}{19} a^{19} - \frac{1}{19} a$, $\frac{1}{19} a^{20} - \frac{1}{19} a^{2}$, $\frac{1}{19} a^{21} - \frac{1}{19} a^{3}$, $\frac{1}{19} a^{22} - \frac{1}{19} a^{4}$, $\frac{1}{361} a^{23} - \frac{3}{361} a^{22} - \frac{9}{361} a^{21} - \frac{2}{361} a^{20} - \frac{6}{361} a^{19} + \frac{3}{361} a^{18} + \frac{2}{361} a^{17} + \frac{3}{361} a^{16} + \frac{6}{361} a^{15} - \frac{7}{361} a^{14} + \frac{8}{361} a^{12} - \frac{23}{361} a^{11} + \frac{12}{361} a^{10} - \frac{18}{361} a^{9} + \frac{78}{361} a^{8} - \frac{71}{361} a^{7} - \frac{50}{361} a^{6} - \frac{8}{19} a^{5} - \frac{89}{361} a^{4} + \frac{153}{361} a^{3} + \frac{44}{361} a^{2} + \frac{6}{19} a$, $\frac{1}{361} a^{24} + \frac{1}{361} a^{22} + \frac{9}{361} a^{21} + \frac{7}{361} a^{20} + \frac{4}{361} a^{19} - \frac{8}{361} a^{18} + \frac{9}{361} a^{17} - \frac{4}{361} a^{16} - \frac{8}{361} a^{15} - \frac{2}{361} a^{14} + \frac{8}{361} a^{13} + \frac{153}{361} a^{12} - \frac{2}{19} a^{11} + \frac{94}{361} a^{10} - \frac{147}{361} a^{9} - \frac{160}{361} a^{8} - \frac{111}{361} a^{7} - \frac{93}{361} a^{6} + \frac{139}{361} a^{5} + \frac{8}{19} a^{4} - \frac{86}{361} a^{3} + \frac{151}{361} a^{2} + \frac{8}{19} a$, $\frac{1}{361} a^{25} - \frac{7}{361} a^{22} - \frac{3}{361} a^{21} + \frac{6}{361} a^{20} - \frac{2}{361} a^{19} + \frac{6}{361} a^{18} - \frac{6}{361} a^{17} + \frac{8}{361} a^{16} - \frac{8}{361} a^{15} - \frac{4}{361} a^{14} + \frac{1}{361} a^{13} - \frac{84}{361} a^{12} + \frac{155}{361} a^{11} + \frac{69}{361} a^{10} - \frac{28}{361} a^{9} - \frac{37}{361} a^{8} - \frac{41}{361} a^{7} + \frac{18}{361} a^{6} + \frac{3}{19} a^{5} - \frac{73}{361} a^{4} + \frac{55}{361} a^{3} + \frac{127}{361} a^{2} - \frac{8}{19} a$, $\frac{1}{6859} a^{26} - \frac{4}{6859} a^{25} - \frac{9}{6859} a^{24} + \frac{7}{6859} a^{23} + \frac{12}{6859} a^{22} + \frac{96}{6859} a^{21} - \frac{79}{6859} a^{20} + \frac{27}{6859} a^{19} + \frac{103}{6859} a^{18} + \frac{112}{6859} a^{17} - \frac{4}{361} a^{16} - \frac{139}{6859} a^{15} + \frac{89}{6859} a^{14} - \frac{141}{6859} a^{13} - \frac{3263}{6859} a^{12} + \frac{533}{6859} a^{11} + \frac{2476}{6859} a^{10} - \frac{3034}{6859} a^{9} + \frac{777}{6859} a^{8} + \frac{567}{6859} a^{7} - \frac{1930}{6859} a^{6} - \frac{2540}{6859} a^{5} + \frac{1381}{6859} a^{4} + \frac{353}{6859} a^{3} - \frac{2448}{6859} a^{2} - \frac{51}{361} a + \frac{9}{19}$, $\frac{1}{3509179895240640535178979475066999358564118859129726175206081} a^{27} - \frac{185337548946174656167982041656717579751892859224038197350}{3509179895240640535178979475066999358564118859129726175206081} a^{26} + \frac{2754872406275483565970006950218990838711173062067554319435}{3509179895240640535178979475066999358564118859129726175206081} a^{25} + \frac{1144206943918431053247498637557069617004006067573549975329}{3509179895240640535178979475066999358564118859129726175206081} a^{24} - \frac{4039329178466550430385706238639036793659195316237751442744}{3509179895240640535178979475066999358564118859129726175206081} a^{23} + \frac{61320491447138222227245831352492687158700188904444403430947}{3509179895240640535178979475066999358564118859129726175206081} a^{22} - \frac{644890822943176639480940658885149127016605919227961130311}{184693678696875817640998919740368387292848361006827693431899} a^{21} + \frac{8012778463826534786055413988727912469402559240543837077260}{3509179895240640535178979475066999358564118859129726175206081} a^{20} + \frac{61781555290201329043241381595649076574089751343114854000534}{3509179895240640535178979475066999358564118859129726175206081} a^{19} + \frac{45060665467278430319400149113782416321333594684028676888352}{3509179895240640535178979475066999358564118859129726175206081} a^{18} + \frac{46891572206491086901484280484496485055778867084823674409508}{3509179895240640535178979475066999358564118859129726175206081} a^{17} + \frac{9861431978873295381412997947475403104518232995609731443410}{3509179895240640535178979475066999358564118859129726175206081} a^{16} - \frac{83808937257838646776043476053569209357455528643895782799455}{3509179895240640535178979475066999358564118859129726175206081} a^{15} + \frac{62814655883233614073930264055556212870092051320807648804441}{3509179895240640535178979475066999358564118859129726175206081} a^{14} - \frac{1963279553203212856732643113439106125401762896266425012768}{184693678696875817640998919740368387292848361006827693431899} a^{13} + \frac{339292903233185240124412807279147041246987941257680411893098}{3509179895240640535178979475066999358564118859129726175206081} a^{12} - \frac{371229731399397493858763180564015517124813077983627460004288}{3509179895240640535178979475066999358564118859129726175206081} a^{11} + \frac{1049643228003622212396873544745943873023201362672826156716423}{3509179895240640535178979475066999358564118859129726175206081} a^{10} + \frac{746659305118330349224054197743455783965258447546143500985771}{3509179895240640535178979475066999358564118859129726175206081} a^{9} - \frac{1226712120985426389896042964231803573373515231582471378806996}{3509179895240640535178979475066999358564118859129726175206081} a^{8} + \frac{469595082163269006212913818968407083438480320369038356632628}{3509179895240640535178979475066999358564118859129726175206081} a^{7} + \frac{533838113154550048464541246312293039688110581065678908488881}{3509179895240640535178979475066999358564118859129726175206081} a^{6} - \frac{532521938836073146505537319687409568180681045867537203175659}{3509179895240640535178979475066999358564118859129726175206081} a^{5} + \frac{1098888875941227404673354867957538492535760036488580301387219}{3509179895240640535178979475066999358564118859129726175206081} a^{4} + \frac{5426138851234963770237122988416673200700889610722269394452}{3509179895240640535178979475066999358564118859129726175206081} a^{3} - \frac{534033514200018458653245840993081509126088431518409134158505}{3509179895240640535178979475066999358564118859129726175206081} a^{2} - \frac{41579852964134738323564841843537030770392704413432761184428}{184693678696875817640998919740368387292848361006827693431899} a - \frac{3320458677852389902066708939593638047697004777853240352364}{9720719931414516717947311565282546699623597947727773338521}$

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:  \( 98168811512269490000 \) (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, 7.7.13841287201.1, 14.14.14967283701606751125078125.1

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 ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ $28$ $28$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{28}$ $28$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}$ $28$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{2}$ $28$ $28$ $28$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{4}$

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