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Properties

Label 30.30.1748547732...8125.1
Degree $30$
Signature $[30, 0]$
Discriminant $5^{15}\cdot 31^{28}$
Root discriminant $55.13$
Ramified primes $5, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{30}$ (as 30T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -8, 148, 112, -3682, -1092, 35532, 8730, -175815, -41272, 510922, 114974, -939544, -199706, 1140241, 224622, -933247, -166292, 518752, 80952, -194877, -25508, 48698, 5047, -7867, -596, 781, 38, -43, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 - 43*x^28 + 38*x^27 + 781*x^26 - 596*x^25 - 7867*x^24 + 5047*x^23 + 48698*x^22 - 25508*x^21 - 194877*x^20 + 80952*x^19 + 518752*x^18 - 166292*x^17 - 933247*x^16 + 224622*x^15 + 1140241*x^14 - 199706*x^13 - 939544*x^12 + 114974*x^11 + 510922*x^10 - 41272*x^9 - 175815*x^8 + 8730*x^7 + 35532*x^6 - 1092*x^5 - 3682*x^4 + 112*x^3 + 148*x^2 - 8*x - 1)
 
gp: K = bnfinit(x^30 - x^29 - 43*x^28 + 38*x^27 + 781*x^26 - 596*x^25 - 7867*x^24 + 5047*x^23 + 48698*x^22 - 25508*x^21 - 194877*x^20 + 80952*x^19 + 518752*x^18 - 166292*x^17 - 933247*x^16 + 224622*x^15 + 1140241*x^14 - 199706*x^13 - 939544*x^12 + 114974*x^11 + 510922*x^10 - 41272*x^9 - 175815*x^8 + 8730*x^7 + 35532*x^6 - 1092*x^5 - 3682*x^4 + 112*x^3 + 148*x^2 - 8*x - 1, 1)
 

Normalized defining polynomial

\( x^{30} - x^{29} - 43 x^{28} + 38 x^{27} + 781 x^{26} - 596 x^{25} - 7867 x^{24} + 5047 x^{23} + 48698 x^{22} - 25508 x^{21} - 194877 x^{20} + 80952 x^{19} + 518752 x^{18} - 166292 x^{17} - 933247 x^{16} + 224622 x^{15} + 1140241 x^{14} - 199706 x^{13} - 939544 x^{12} + 114974 x^{11} + 510922 x^{10} - 41272 x^{9} - 175815 x^{8} + 8730 x^{7} + 35532 x^{6} - 1092 x^{5} - 3682 x^{4} + 112 x^{3} + 148 x^{2} - 8 x - 1 \)

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

Invariants

Degree:  $30$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[30, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17485477327500765872904889567178150559785186767578125=5^{15}\cdot 31^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.13$
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$$\chi_{155}(64,·)$, $\chi_{155}(1,·)$, $\chi_{155}(66,·)$, $\chi_{155}(131,·)$, $\chi_{155}(4,·)$, $\chi_{155}(69,·)$, $\chi_{155}(134,·)$, $\chi_{155}(129,·)$, $\chi_{155}(9,·)$, $\chi_{155}(76,·)$, $\chi_{155}(14,·)$, $\chi_{155}(16,·)$, $\chi_{155}(81,·)$, $\chi_{155}(19,·)$, $\chi_{155}(149,·)$, $\chi_{155}(94,·)$, $\chi_{155}(144,·)$, $\chi_{155}(36,·)$, $\chi_{155}(101,·)$, $\chi_{155}(39,·)$, $\chi_{155}(41,·)$, $\chi_{155}(71,·)$, $\chi_{155}(109,·)$, $\chi_{155}(111,·)$, $\chi_{155}(49,·)$, $\chi_{155}(51,·)$, $\chi_{155}(56,·)$, $\chi_{155}(121,·)$, $\chi_{155}(59,·)$, $\chi_{155}(126,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{61} a^{27} + \frac{3}{61} a^{26} - \frac{23}{61} a^{25} + \frac{23}{61} a^{24} - \frac{6}{61} a^{23} - \frac{8}{61} a^{22} - \frac{18}{61} a^{21} + \frac{18}{61} a^{20} + \frac{12}{61} a^{19} + \frac{21}{61} a^{18} - \frac{7}{61} a^{17} - \frac{12}{61} a^{16} - \frac{20}{61} a^{15} - \frac{4}{61} a^{14} - \frac{27}{61} a^{13} - \frac{21}{61} a^{12} + \frac{5}{61} a^{11} + \frac{15}{61} a^{10} + \frac{2}{61} a^{9} + \frac{16}{61} a^{8} + \frac{7}{61} a^{7} - \frac{18}{61} a^{6} + \frac{26}{61} a^{5} - \frac{28}{61} a^{4} + \frac{26}{61} a^{3} - \frac{17}{61} a^{2} - \frac{24}{61} a - \frac{23}{61}$, $\frac{1}{61} a^{28} + \frac{29}{61} a^{26} - \frac{30}{61} a^{25} - \frac{14}{61} a^{24} + \frac{10}{61} a^{23} + \frac{6}{61} a^{22} + \frac{11}{61} a^{21} + \frac{19}{61} a^{20} - \frac{15}{61} a^{19} - \frac{9}{61} a^{18} + \frac{9}{61} a^{17} + \frac{16}{61} a^{16} - \frac{5}{61} a^{15} - \frac{15}{61} a^{14} - \frac{1}{61} a^{13} + \frac{7}{61} a^{12} + \frac{18}{61} a^{10} + \frac{10}{61} a^{9} + \frac{20}{61} a^{8} + \frac{22}{61} a^{7} + \frac{19}{61} a^{6} + \frac{16}{61} a^{5} - \frac{12}{61} a^{4} + \frac{27}{61} a^{3} + \frac{27}{61} a^{2} - \frac{12}{61} a + \frac{8}{61}$, $\frac{1}{2115390618211016903709730225178783779} a^{29} - \frac{3914262553796229550043625838232999}{2115390618211016903709730225178783779} a^{28} + \frac{15566156993351725186920753418952928}{2115390618211016903709730225178783779} a^{27} - \frac{912460526940827941093716855000582118}{2115390618211016903709730225178783779} a^{26} - \frac{37166894967273828363785758253769569}{2115390618211016903709730225178783779} a^{25} - \frac{137251088615936538764383287477502236}{2115390618211016903709730225178783779} a^{24} + \frac{610756932690000571996235671633328654}{2115390618211016903709730225178783779} a^{23} - \frac{595876227819456012988960489243886989}{2115390618211016903709730225178783779} a^{22} - \frac{254014670709973643542797202298356001}{2115390618211016903709730225178783779} a^{21} + \frac{323139794954024563351727434178859496}{2115390618211016903709730225178783779} a^{20} - \frac{796591076070051754790417924535090360}{2115390618211016903709730225178783779} a^{19} - \frac{299608568478163929947632976925983603}{2115390618211016903709730225178783779} a^{18} + \frac{181435910439287600168047888899456503}{2115390618211016903709730225178783779} a^{17} - \frac{953925232744836993653524269888682236}{2115390618211016903709730225178783779} a^{16} + \frac{483120491973077363629537239407551847}{2115390618211016903709730225178783779} a^{15} - \frac{142768592371530831724507553160530136}{2115390618211016903709730225178783779} a^{14} + \frac{125077344938463234770308728785972797}{2115390618211016903709730225178783779} a^{13} + \frac{757477035320596550285199883029555435}{2115390618211016903709730225178783779} a^{12} + \frac{620016483901452179935669336347124230}{2115390618211016903709730225178783779} a^{11} - \frac{946106994638078726400775394093384378}{2115390618211016903709730225178783779} a^{10} + \frac{271963025026299094036865022193425854}{2115390618211016903709730225178783779} a^{9} - \frac{782365499971169656755918094600939317}{2115390618211016903709730225178783779} a^{8} - \frac{412127257412285521228411975584540736}{2115390618211016903709730225178783779} a^{7} + \frac{721524124483155870890769650172857519}{2115390618211016903709730225178783779} a^{6} + \frac{1031197464034190528322021885208302501}{2115390618211016903709730225178783779} a^{5} + \frac{278254308699982268350503993380858038}{2115390618211016903709730225178783779} a^{4} - \frac{479990593183451080422874155901123699}{2115390618211016903709730225178783779} a^{3} - \frac{3176727738794442955450571061151495}{34678534724770768913274265986537439} a^{2} + \frac{942885208220860925047650030959937234}{2115390618211016903709730225178783779} a + \frac{815797806870753123022079888263044792}{2115390618211016903709730225178783779}$

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:  $29$
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:  \( 43214123905561144 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), 3.3.961.1, 5.5.923521.1, 6.6.115440125.1, 10.10.2665284492003125.1, \(\Q(\zeta_{31})^+\)

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{3}$ $30$ R $30$ $15^{2}$ $30$ $30$ $15^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{6}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{5}$ $15^{2}$ $30$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{3}$ $30$ $15^{2}$

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$$[\ ]_{2}^{3}$
$31$31.15.14.1$x^{15} - 31$$15$$1$$14$$C_{15}$$[\ ]_{15}$
31.15.14.1$x^{15} - 31$$15$$1$$14$$C_{15}$$[\ ]_{15}$