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Properties

Label 30.0.12858032007...7184.1
Degree $30$
Signature $[0, 15]$
Discriminant $-\,2^{30}\cdot 3^{40}\cdot 11^{24}$
Root discriminant $58.93$
Ramified primes $2, 3, 11$
Class number $19231$ (GRH)
Class group $[19231]$ (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, 0, 135, 0, 3765, 0, 46110, 0, 305550, 0, 1189389, 0, 2798498, 0, 3980655, 0, 3399342, 0, 1774210, 0, 578880, 0, 119634, 0, 15576, 0, 1233, 0, 54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^30 + 54*x^28 + 1233*x^26 + 15576*x^24 + 119634*x^22 + 578880*x^20 + 1774210*x^18 + 3399342*x^16 + 3980655*x^14 + 2798498*x^12 + 1189389*x^10 + 305550*x^8 + 46110*x^6 + 3765*x^4 + 135*x^2 + 1)
 
gp: K = bnfinit(x^30 + 54*x^28 + 1233*x^26 + 15576*x^24 + 119634*x^22 + 578880*x^20 + 1774210*x^18 + 3399342*x^16 + 3980655*x^14 + 2798498*x^12 + 1189389*x^10 + 305550*x^8 + 46110*x^6 + 3765*x^4 + 135*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{30} + 54 x^{28} + 1233 x^{26} + 15576 x^{24} + 119634 x^{22} + 578880 x^{20} + 1774210 x^{18} + 3399342 x^{16} + 3980655 x^{14} + 2798498 x^{12} + 1189389 x^{10} + 305550 x^{8} + 46110 x^{6} + 3765 x^{4} + 135 x^{2} + 1 \)

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

Invariants

Degree:  $30$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 15]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-128580320071219649477610441499385131775609876653277184=-\,2^{30}\cdot 3^{40}\cdot 11^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.93$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(396=2^{2}\cdot 3^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{396}(1,·)$, $\chi_{396}(67,·)$, $\chi_{396}(97,·)$, $\chi_{396}(133,·)$, $\chi_{396}(199,·)$, $\chi_{396}(265,·)$, $\chi_{396}(331,·)$, $\chi_{396}(355,·)$, $\chi_{396}(235,·)$, $\chi_{396}(25,·)$, $\chi_{396}(91,·)$, $\chi_{396}(157,·)$, $\chi_{396}(31,·)$, $\chi_{396}(289,·)$, $\chi_{396}(163,·)$, $\chi_{396}(37,·)$, $\chi_{396}(295,·)$, $\chi_{396}(379,·)$, $\chi_{396}(169,·)$, $\chi_{396}(103,·)$, $\chi_{396}(301,·)$, $\chi_{396}(367,·)$, $\chi_{396}(229,·)$, $\chi_{396}(49,·)$, $\chi_{396}(115,·)$, $\chi_{396}(181,·)$, $\chi_{396}(361,·)$, $\chi_{396}(313,·)$, $\chi_{396}(223,·)$, $\chi_{396}(247,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{109} a^{26} - \frac{15}{109} a^{24} + \frac{15}{109} a^{22} + \frac{49}{109} a^{20} + \frac{54}{109} a^{18} - \frac{19}{109} a^{16} + \frac{2}{109} a^{14} + \frac{9}{109} a^{12} - \frac{29}{109} a^{10} - \frac{41}{109} a^{8} + \frac{22}{109} a^{6} - \frac{28}{109} a^{4} + \frac{2}{109} a^{2} + \frac{3}{109}$, $\frac{1}{109} a^{27} - \frac{15}{109} a^{25} + \frac{15}{109} a^{23} + \frac{49}{109} a^{21} + \frac{54}{109} a^{19} - \frac{19}{109} a^{17} + \frac{2}{109} a^{15} + \frac{9}{109} a^{13} - \frac{29}{109} a^{11} - \frac{41}{109} a^{9} + \frac{22}{109} a^{7} - \frac{28}{109} a^{5} + \frac{2}{109} a^{3} + \frac{3}{109} a$, $\frac{1}{20469756790058585157336583} a^{28} - \frac{34572278632687996029752}{20469756790058585157336583} a^{26} + \frac{5600567293310221981624488}{20469756790058585157336583} a^{24} + \frac{9995868499639387490619561}{20469756790058585157336583} a^{22} - \frac{7452536763467663301093690}{20469756790058585157336583} a^{20} - \frac{8293010612449553670769619}{20469756790058585157336583} a^{18} + \frac{2257724728493536047787222}{20469756790058585157336583} a^{16} - \frac{164141650495050133016284}{20469756790058585157336583} a^{14} + \frac{4896578976157513160191002}{20469756790058585157336583} a^{12} - \frac{99880876764923358134437}{20469756790058585157336583} a^{10} + \frac{3770786365144958276585776}{20469756790058585157336583} a^{8} - \frac{8721738970238176741444724}{20469756790058585157336583} a^{6} + \frac{10144043949885991434295548}{20469756790058585157336583} a^{4} - \frac{2301635129904606018402975}{20469756790058585157336583} a^{2} - \frac{267434378538790360427201}{20469756790058585157336583}$, $\frac{1}{20469756790058585157336583} a^{29} - \frac{34572278632687996029752}{20469756790058585157336583} a^{27} + \frac{5600567293310221981624488}{20469756790058585157336583} a^{25} + \frac{9995868499639387490619561}{20469756790058585157336583} a^{23} - \frac{7452536763467663301093690}{20469756790058585157336583} a^{21} - \frac{8293010612449553670769619}{20469756790058585157336583} a^{19} + \frac{2257724728493536047787222}{20469756790058585157336583} a^{17} - \frac{164141650495050133016284}{20469756790058585157336583} a^{15} + \frac{4896578976157513160191002}{20469756790058585157336583} a^{13} - \frac{99880876764923358134437}{20469756790058585157336583} a^{11} + \frac{3770786365144958276585776}{20469756790058585157336583} a^{9} - \frac{8721738970238176741444724}{20469756790058585157336583} a^{7} + \frac{10144043949885991434295548}{20469756790058585157336583} a^{5} - \frac{2301635129904606018402975}{20469756790058585157336583} a^{3} - \frac{267434378538790360427201}{20469756790058585157336583} a$

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

Class group and class number

$C_{19231}$, which has order $19231$ (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:  $14$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{11340062264187261}{13582021966880203} a^{29} - \frac{607907297716553567}{13582021966880203} a^{27} - \frac{13742838082729367400}{13582021966880203} a^{25} - \frac{171201334936926036060}{13582021966880203} a^{23} - \frac{1288676355534596912938}{13582021966880203} a^{21} - \frac{6049322690772542026674}{13582021966880203} a^{19} - \frac{17675924983237737788580}{13582021966880203} a^{17} - \frac{31288695028267565504478}{13582021966880203} a^{15} - \frac{31920602521404356206692}{13582021966880203} a^{13} - \frac{17531623726854618720627}{13582021966880203} a^{11} - \frac{4859609699009417524696}{13582021966880203} a^{9} - \frac{544374694989261541302}{13582021966880203} a^{7} + \frac{10380630056743394109}{13582021966880203} a^{5} + \frac{5388374271851591333}{13582021966880203} a^{3} + \frac{188595294201069804}{13582021966880203} a \) (order $4$)
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:  \( 15853905121.091976 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{11})^+\), 6.0.419904.1, 10.0.219503494144.1, 15.15.10943023107606534329121.1

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 R R $15^{2}$ $30$ R $15^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ $15^{2}$ $30$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{6}$ $15^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{5}$ $30$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{6}$ $30$

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
2Data not computed
3Data not computed
11Data not computed