↑

Properties

Label 30.0.54204979715...1875.1
Degree $30$
Signature $[0, 15]$
Discriminant $-\,5^{15}\cdot 31^{29}$
Root discriminant $61.82$
Ramified primes $5, 31$
Class number Not computed
Class group Not computed
Galois group $C_{30}$ (as 30T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3010349, -3010318, 3010318, -3009078, 3009078, -2994322, 2994322, -2912110, 2912110, -2651772, 2651772, -2131096, 2131096, -1430186, 1430186, -769328, 769328, -322277, 322277, -102673, 102673, -24243, 24243, -4093, 4093, -466, 466, -32, 32, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 + 32*x^28 - 32*x^27 + 466*x^26 - 466*x^25 + 4093*x^24 - 4093*x^23 + 24243*x^22 - 24243*x^21 + 102673*x^20 - 102673*x^19 + 322277*x^18 - 322277*x^17 + 769328*x^16 - 769328*x^15 + 1430186*x^14 - 1430186*x^13 + 2131096*x^12 - 2131096*x^11 + 2651772*x^10 - 2651772*x^9 + 2912110*x^8 - 2912110*x^7 + 2994322*x^6 - 2994322*x^5 + 3009078*x^4 - 3009078*x^3 + 3010318*x^2 - 3010318*x + 3010349)
 
gp: K = bnfinit(x^30 - x^29 + 32*x^28 - 32*x^27 + 466*x^26 - 466*x^25 + 4093*x^24 - 4093*x^23 + 24243*x^22 - 24243*x^21 + 102673*x^20 - 102673*x^19 + 322277*x^18 - 322277*x^17 + 769328*x^16 - 769328*x^15 + 1430186*x^14 - 1430186*x^13 + 2131096*x^12 - 2131096*x^11 + 2651772*x^10 - 2651772*x^9 + 2912110*x^8 - 2912110*x^7 + 2994322*x^6 - 2994322*x^5 + 3009078*x^4 - 3009078*x^3 + 3010318*x^2 - 3010318*x + 3010349, 1)
 

Normalized defining polynomial

\( x^{30} - x^{29} + 32 x^{28} - 32 x^{27} + 466 x^{26} - 466 x^{25} + 4093 x^{24} - 4093 x^{23} + 24243 x^{22} - 24243 x^{21} + 102673 x^{20} - 102673 x^{19} + 322277 x^{18} - 322277 x^{17} + 769328 x^{16} - 769328 x^{15} + 1430186 x^{14} - 1430186 x^{13} + 2131096 x^{12} - 2131096 x^{11} + 2651772 x^{10} - 2651772 x^{9} + 2912110 x^{8} - 2912110 x^{7} + 2994322 x^{6} - 2994322 x^{5} + 3009078 x^{4} - 3009078 x^{3} + 3010318 x^{2} - 3010318 x + 3010349 \)

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

Invariants

Degree:  $30$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 15]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-542049797152523742060051576582522667353340789794921875=-\,5^{15}\cdot 31^{29}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.82$
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:  $5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(155=5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{155}(1,·)$, $\chi_{155}(66,·)$, $\chi_{155}(131,·)$, $\chi_{155}(71,·)$, $\chi_{155}(74,·)$, $\chi_{155}(139,·)$, $\chi_{155}(76,·)$, $\chi_{155}(79,·)$, $\chi_{155}(16,·)$, $\chi_{155}(81,·)$, $\chi_{155}(84,·)$, $\chi_{155}(24,·)$, $\chi_{155}(89,·)$, $\chi_{155}(154,·)$, $\chi_{155}(29,·)$, $\chi_{155}(34,·)$, $\chi_{155}(99,·)$, $\chi_{155}(36,·)$, $\chi_{155}(101,·)$, $\chi_{155}(104,·)$, $\chi_{155}(41,·)$, $\chi_{155}(44,·)$, $\chi_{155}(111,·)$, $\chi_{155}(114,·)$, $\chi_{155}(51,·)$, $\chi_{155}(54,·)$, $\chi_{155}(119,·)$, $\chi_{155}(56,·)$, $\chi_{155}(121,·)$, $\chi_{155}(126,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{1346269} a^{16} + \frac{514229}{1346269} a^{15} + \frac{16}{1346269} a^{14} - \frac{364179}{1346269} a^{13} + \frac{104}{1346269} a^{12} + \frac{507464}{1346269} a^{11} + \frac{352}{1346269} a^{10} + \frac{54730}{1346269} a^{9} + \frac{660}{1346269} a^{8} - \frac{155218}{1346269} a^{7} + \frac{672}{1346269} a^{6} + \frac{515826}{1346269} a^{5} + \frac{336}{1346269} a^{4} + \frac{639803}{1346269} a^{3} + \frac{64}{1346269} a^{2} - \frac{364179}{1346269} a + \frac{2}{1346269}$, $\frac{1}{1346269} a^{17} + \frac{17}{1346269} a^{15} - \frac{514229}{1346269} a^{14} + \frac{119}{1346269} a^{13} - \frac{467861}{1346269} a^{12} + \frac{442}{1346269} a^{11} - \frac{553832}{1346269} a^{10} + \frac{935}{1346269} a^{9} - \frac{286570}{1346269} a^{8} + \frac{1122}{1346269} a^{7} - \frac{401198}{1346269} a^{6} + \frac{714}{1346269} a^{5} + \frac{181291}{1346269} a^{4} + \frac{204}{1346269} a^{3} + \frac{381890}{1346269} a^{2} + \frac{17}{1346269} a + \frac{317811}{1346269}$, $\frac{1}{1346269} a^{18} + \frac{167761}{1346269} a^{15} - \frac{153}{1346269} a^{14} + \frac{338106}{1346269} a^{13} - \frac{1326}{1346269} a^{12} + \frac{243163}{1346269} a^{11} - \frac{5049}{1346269} a^{10} + \frac{129289}{1346269} a^{9} - \frac{10098}{1346269} a^{8} - \frac{455030}{1346269} a^{7} - \frac{10710}{1346269} a^{6} - \frac{510137}{1346269} a^{5} - \frac{5508}{1346269} a^{4} + \frac{275391}{1346269} a^{3} - \frac{1071}{1346269} a^{2} - \frac{222491}{1346269} a - \frac{34}{1346269}$, $\frac{1}{1346269} a^{19} - \frac{171}{1346269} a^{15} + \frac{346468}{1346269} a^{14} - \frac{1596}{1346269} a^{13} + \frac{297516}{1346269} a^{12} - \frac{6669}{1346269} a^{11} + \frac{313253}{1346269} a^{10} - \frac{15048}{1346269} a^{9} + \frac{563037}{1346269} a^{8} - \frac{18810}{1346269} a^{7} - \frac{158933}{1346269} a^{6} - \frac{12312}{1346269} a^{5} + \frac{450993}{1346269} a^{4} - \frac{3591}{1346269} a^{3} - \frac{189043}{1346269} a^{2} - \frac{304}{1346269} a - \frac{335522}{1346269}$, $\frac{1}{1346269} a^{20} - \frac{574127}{1346269} a^{15} + \frac{1140}{1346269} a^{14} - \frac{48719}{1346269} a^{13} + \frac{11115}{1346269} a^{12} - \frac{417888}{1346269} a^{11} + \frac{45144}{1346269} a^{10} + \frac{497984}{1346269} a^{9} + \frac{94050}{1346269} a^{8} + \frac{224169}{1346269} a^{7} + \frac{102600}{1346269} a^{6} - \frac{196515}{1346269} a^{5} + \frac{53865}{1346269} a^{4} + \frac{169481}{1346269} a^{3} + \frac{10640}{1346269} a^{2} + \frac{664512}{1346269} a + \frac{342}{1346269}$, $\frac{1}{1346269} a^{21} + \frac{1330}{1346269} a^{15} - \frac{286570}{1346269} a^{14} + \frac{13965}{1346269} a^{13} + \frac{55484}{1346269} a^{12} + \frac{62244}{1346269} a^{11} + \frac{650338}{1346269} a^{10} + \frac{146300}{1346269} a^{9} - \frac{499869}{1346269} a^{8} + \frac{188100}{1346269} a^{7} + \frac{583895}{1346269} a^{6} + \frac{125685}{1346269} a^{5} + \frac{559686}{1346269} a^{4} + \frac{37240}{1346269} a^{3} - \frac{286892}{1346269} a^{2} + \frac{3192}{1346269} a - \frac{198015}{1346269}$, $\frac{1}{1346269} a^{22} - \frac{306488}{1346269} a^{15} - \frac{7315}{1346269} a^{14} - \frac{243286}{1346269} a^{13} - \frac{76076}{1346269} a^{12} + \frac{203987}{1346269} a^{11} - \frac{321860}{1346269} a^{10} - \frac{592243}{1346269} a^{9} + \frac{656569}{1346269} a^{8} - \frac{301591}{1346269} a^{7} + \frac{578194}{1346269} a^{6} - \frac{237973}{1346269} a^{5} - \frac{409640}{1346269} a^{4} - \frac{382874}{1346269} a^{3} - \frac{81928}{1346269} a^{2} - \frac{496785}{1346269} a - \frac{2660}{1346269}$, $\frac{1}{1346269} a^{23} - \frac{8855}{1346269} a^{15} + \frac{621715}{1346269} a^{14} - \frac{99176}{1346269} a^{13} - \frac{231717}{1346269} a^{12} - \frac{460460}{1346269} a^{11} - \frac{409987}{1346269} a^{10} + \frac{233069}{1346269} a^{9} + \frac{40139}{1346269} a^{8} - \frac{114806}{1346269} a^{7} - \frac{257194}{1346269} a^{6} + \frac{354509}{1346269} a^{5} + \frac{280650}{1346269} a^{4} - \frac{297528}{1346269} a^{3} + \frac{270681}{1346269} a^{2} - \frac{25760}{1346269} a + \frac{612976}{1346269}$, $\frac{1}{1346269} a^{24} - \frac{308517}{1346269} a^{15} + \frac{42504}{1346269} a^{14} + \frac{623762}{1346269} a^{13} + \frac{460460}{1346269} a^{12} - \frac{662189}{1346269} a^{11} + \frac{657491}{1346269} a^{10} + \frac{17449}{1346269} a^{9} + \frac{344418}{1346269} a^{8} - \frac{171935}{1346269} a^{7} - \frac{426276}{1346269} a^{6} + \frac{29163}{1346269} a^{5} - \frac{14786}{1346269} a^{4} + \frac{626294}{1346269} a^{3} + \frac{540960}{1346269} a^{2} + \frac{122186}{1346269} a + \frac{17710}{1346269}$, $\frac{1}{1346269} a^{25} + \frac{53130}{1346269} a^{15} + \frac{174958}{1346269} a^{14} + \frac{619850}{1346269} a^{13} + \frac{459392}{1346269} a^{12} + \frac{267562}{1346269} a^{11} - \frac{432356}{1346269} a^{10} + \frac{574030}{1346269} a^{9} + \frac{162666}{1346269} a^{8} + \frac{316617}{1346269} a^{7} + \frac{27161}{1346269} a^{6} - \frac{36965}{1346269} a^{5} + \frac{625293}{1346269} a^{4} - \frac{663938}{1346269} a^{3} - \frac{326761}{1346269} a^{2} + \frac{177100}{1346269} a + \frac{617034}{1346269}$, $\frac{1}{1346269} a^{26} + \frac{371274}{1346269} a^{15} - \frac{230230}{1346269} a^{14} - \frac{634675}{1346269} a^{13} + \frac{127118}{1346269} a^{12} - \frac{265413}{1346269} a^{11} - \frac{626233}{1346269} a^{10} + \frac{298806}{1346269} a^{9} + \frac{253811}{1346269} a^{8} - \frac{484393}{1346269} a^{7} + \frac{608938}{1346269} a^{6} - \frac{558323}{1346269} a^{5} + \frac{332148}{1346269} a^{4} + \frac{232099}{1346269} a^{3} - \frac{530682}{1346269} a^{2} - \frac{477033}{1346269} a - \frac{106260}{1346269}$, $\frac{1}{1346269} a^{27} - \frac{296010}{1346269} a^{15} + \frac{156286}{1346269} a^{14} + \frac{486687}{1346269} a^{13} + \frac{163892}{1346269} a^{12} + \frac{184912}{1346269} a^{11} + \frac{198451}{1346269} a^{10} - \frac{334192}{1346269} a^{9} - \frac{504275}{1346269} a^{8} + \frac{625856}{1346269} a^{7} + \frac{351583}{1346269} a^{6} - \frac{299850}{1346269} a^{5} - \frac{659217}{1346269} a^{4} - \frac{315999}{1346269} a^{3} - \frac{5727}{1346269} a^{2} + \frac{253309}{1346269} a + \frac{603721}{1346269}$, $\frac{1}{1346269} a^{28} - \frac{168178}{1346269} a^{15} - \frac{162229}{1346269} a^{14} - \frac{664261}{1346269} a^{13} + \frac{5765}{1346269} a^{12} + \frac{614609}{1346269} a^{11} + \frac{198615}{1346269} a^{10} + \frac{468148}{1346269} a^{9} - \frac{562818}{1346269} a^{8} - \frac{260165}{1346269} a^{7} - \frac{628942}{1346269} a^{6} + \frac{550139}{1346269} a^{5} - \frac{480545}{1346269} a^{4} + \frac{342459}{1346269} a^{3} + \frac{350183}{1346269} a^{2} - \frac{224432}{1346269} a + \frac{592020}{1346269}$, $\frac{1}{1346269} a^{29} + \frac{214511}{1346269} a^{15} - \frac{665951}{1346269} a^{14} + \frac{271789}{1346269} a^{13} + \frac{603624}{1346269} a^{12} + \frac{448490}{1346269} a^{11} + \frac{430968}{1346269} a^{10} - \frac{622031}{1346269} a^{9} + \frac{343257}{1346269} a^{8} + \frac{620433}{1346269} a^{7} + \frac{479159}{1346269} a^{6} + \frac{568930}{1346269} a^{5} + \frac{306969}{1346269} a^{4} + \frac{589292}{1346269} a^{3} - \frac{231192}{1346269} a^{2} - \frac{488225}{1346269} a + \frac{336356}{1346269}$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $14$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{30}$ (as 30T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 30
The 30 conjugacy class representatives for $C_{30}$
Character table for $C_{30}$ is not computed

Intermediate fields

\(\Q(\sqrt{-155}) \), 3.3.961.1, 5.5.923521.1, 6.0.3578643875.1, 10.0.82623819252096875.1, \(\Q(\zeta_{31})^+\)

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{3}$ $15^{2}$ R $30$ $30$ $15^{2}$ $15^{2}$ $15^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{10}$ $15^{2}$ $15^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{3}$ $15^{2}$ $15^{2}$

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
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31Data not computed