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Properties

Label 30.0.20099782588...7184.1
Degree $30$
Signature $[0, 15]$
Discriminant $-\,2^{30}\cdot 11^{24}\cdot 13^{20}$
Root discriminant $75.30$
Ramified primes $2, 11, 13$
Class number $82893$ (GRH)
Class group $[82893]$ (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![17161, 0, 428115, 0, 4040689, 0, 19688626, 0, 55508682, 0, 95476739, 0, 103666360, 0, 73408942, 0, 34753087, 0, 11154782, 0, 2434242, 0, 357772, 0, 34524, 0, 2077, 0, 70, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^30 + 70*x^28 + 2077*x^26 + 34524*x^24 + 357772*x^22 + 2434242*x^20 + 11154782*x^18 + 34753087*x^16 + 73408942*x^14 + 103666360*x^12 + 95476739*x^10 + 55508682*x^8 + 19688626*x^6 + 4040689*x^4 + 428115*x^2 + 17161)
 
gp: K = bnfinit(x^30 + 70*x^28 + 2077*x^26 + 34524*x^24 + 357772*x^22 + 2434242*x^20 + 11154782*x^18 + 34753087*x^16 + 73408942*x^14 + 103666360*x^12 + 95476739*x^10 + 55508682*x^8 + 19688626*x^6 + 4040689*x^4 + 428115*x^2 + 17161, 1)
 

Normalized defining polynomial

\( x^{30} + 70 x^{28} + 2077 x^{26} + 34524 x^{24} + 357772 x^{22} + 2434242 x^{20} + 11154782 x^{18} + 34753087 x^{16} + 73408942 x^{14} + 103666360 x^{12} + 95476739 x^{10} + 55508682 x^{8} + 19688626 x^{6} + 4040689 x^{4} + 428115 x^{2} + 17161 \)

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:  \(-200997825885699545538652088865450292514220326691153117184=-\,2^{30}\cdot 11^{24}\cdot 13^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.30$
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:  $2, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(572=2^{2}\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{572}(1,·)$, $\chi_{572}(3,·)$, $\chi_{572}(133,·)$, $\chi_{572}(81,·)$, $\chi_{572}(9,·)$, $\chi_{572}(471,·)$, $\chi_{572}(269,·)$, $\chi_{572}(399,·)$, $\chi_{572}(529,·)$, $\chi_{572}(339,·)$, $\chi_{572}(313,·)$, $\chi_{572}(27,·)$, $\chi_{572}(157,·)$, $\chi_{572}(287,·)$, $\chi_{572}(367,·)$, $\chi_{572}(289,·)$, $\chi_{572}(419,·)$, $\chi_{572}(295,·)$, $\chi_{572}(555,·)$, $\chi_{572}(477,·)$, $\chi_{572}(113,·)$, $\chi_{572}(443,·)$, $\chi_{572}(243,·)$, $\chi_{572}(235,·)$, $\chi_{572}(53,·)$, $\chi_{572}(521,·)$, $\chi_{572}(185,·)$, $\chi_{572}(159,·)$, $\chi_{572}(445,·)$, $\chi_{572}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{463} a^{22} + \frac{176}{463} a^{20} + \frac{152}{463} a^{18} - \frac{96}{463} a^{16} - \frac{142}{463} a^{14} + \frac{66}{463} a^{12} - \frac{52}{463} a^{10} + \frac{227}{463} a^{8} - \frac{174}{463} a^{6} - \frac{143}{463} a^{4} + \frac{32}{463} a^{2} - \frac{183}{463}$, $\frac{1}{463} a^{23} + \frac{176}{463} a^{21} + \frac{152}{463} a^{19} - \frac{96}{463} a^{17} - \frac{142}{463} a^{15} + \frac{66}{463} a^{13} - \frac{52}{463} a^{11} + \frac{227}{463} a^{9} - \frac{174}{463} a^{7} - \frac{143}{463} a^{5} + \frac{32}{463} a^{3} - \frac{183}{463} a$, $\frac{1}{50467} a^{24} + \frac{54}{50467} a^{22} + \frac{18035}{50467} a^{20} - \frac{21881}{50467} a^{18} + \frac{458}{50467} a^{16} - \frac{2519}{50467} a^{14} + \frac{22917}{50467} a^{12} - \frac{4541}{50467} a^{10} + \frac{838}{50467} a^{8} + \frac{13677}{50467} a^{6} - \frac{21414}{50467} a^{4} - \frac{11495}{50467} a^{2} + \frac{16770}{50467}$, $\frac{1}{50467} a^{25} + \frac{54}{50467} a^{23} + \frac{18035}{50467} a^{21} - \frac{21881}{50467} a^{19} + \frac{458}{50467} a^{17} - \frac{2519}{50467} a^{15} + \frac{22917}{50467} a^{13} - \frac{4541}{50467} a^{11} + \frac{838}{50467} a^{9} + \frac{13677}{50467} a^{7} - \frac{21414}{50467} a^{5} - \frac{11495}{50467} a^{3} + \frac{16770}{50467} a$, $\frac{1}{50467} a^{26} - \frac{32}{50467} a^{22} + \frac{21744}{50467} a^{20} - \frac{10646}{50467} a^{18} + \frac{14169}{50467} a^{16} - \frac{11097}{50467} a^{14} - \frac{21477}{50467} a^{12} + \frac{24564}{50467} a^{10} + \frac{11371}{50467} a^{8} + \frac{9023}{50467} a^{6} - \frac{19368}{50467} a^{4} + \frac{1267}{50467} a^{2} - \frac{226}{50467}$, $\frac{1}{6611177} a^{27} - \frac{42}{6611177} a^{25} - \frac{1210}{6611177} a^{23} - \frac{1401825}{6611177} a^{21} - \frac{742776}{6611177} a^{19} - \frac{3087260}{6611177} a^{17} - \frac{867551}{6611177} a^{15} - \frac{508315}{6611177} a^{13} + \frac{1975418}{6611177} a^{11} - \frac{735268}{6611177} a^{9} - \frac{1158807}{6611177} a^{7} + \frac{370881}{6611177} a^{5} - \frac{843672}{6611177} a^{3} + \frac{408106}{6611177} a$, $\frac{1}{100239256236658258593876419} a^{28} + \frac{273065104455502171323}{100239256236658258593876419} a^{26} + \frac{34554053359108253684}{100239256236658258593876419} a^{24} - \frac{54254522271295184272599}{100239256236658258593876419} a^{22} + \frac{6802938048994185768021841}{100239256236658258593876419} a^{20} - \frac{11082507291005147424007157}{100239256236658258593876419} a^{18} - \frac{38268232183370439073963774}{100239256236658258593876419} a^{16} - \frac{24478393254562750615248308}{100239256236658258593876419} a^{14} - \frac{33730765517414664684825720}{100239256236658258593876419} a^{12} - \frac{45489097467099949267526733}{100239256236658258593876419} a^{10} + \frac{32915916123579430253314986}{100239256236658258593876419} a^{8} + \frac{21137200046469820444703225}{100239256236658258593876419} a^{6} + \frac{23824928273807358845798647}{100239256236658258593876419} a^{4} + \frac{1006019626785523569509812}{100239256236658258593876419} a^{2} - \frac{195765518155017651734868}{765185162111895103770049}$, $\frac{1}{100239256236658258593876419} a^{29} + \frac{147496553029033077}{100239256236658258593876419} a^{27} - \frac{420308626478346976726}{100239256236658258593876419} a^{25} + \frac{65435010883267202022619}{100239256236658258593876419} a^{23} - \frac{50054639175169143475264260}{100239256236658258593876419} a^{21} + \frac{17018378396226359679620498}{100239256236658258593876419} a^{19} - \frac{44640737031178563530613098}{100239256236658258593876419} a^{17} - \frac{19652876380881811397223394}{100239256236658258593876419} a^{15} - \frac{39296571402531062290342909}{100239256236658258593876419} a^{13} + \frac{48421062983496006013855823}{100239256236658258593876419} a^{11} + \frac{21169709062439593450124963}{100239256236658258593876419} a^{9} - \frac{1179348562162047645543930}{100239256236658258593876419} a^{7} + \frac{15646118238096715923668172}{100239256236658258593876419} a^{5} - \frac{18851556496734389460315230}{100239256236658258593876419} a^{3} - \frac{15161213619091118432245006}{100239256236658258593876419} a$

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

Class group and class number

$C_{82893}$, which has order $82893$ (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:  \( -\frac{20264186859181}{28886420612587} a^{29} - \frac{1412206980484869}{28886420612587} a^{27} - \frac{41650640010959341}{28886420612587} a^{25} - \frac{686680513155750989}{28886420612587} a^{23} - \frac{7036947063150442136}{28886420612587} a^{21} - \frac{359885876854603119}{220507027577} a^{19} - \frac{211418118908091960408}{28886420612587} a^{17} - \frac{638661324861810648144}{28886420612587} a^{15} - \frac{1289462304298929655434}{28886420612587} a^{13} - \frac{1700733723430232766249}{28886420612587} a^{11} - \frac{1407214205458401532778}{28886420612587} a^{9} - \frac{688350935750064354818}{28886420612587} a^{7} - \frac{185470409763784403928}{28886420612587} a^{5} - \frac{24357506181722527341}{28886420612587} a^{3} - \frac{1121450382722044276}{28886420612587} a \) (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:  \( 85915831770.81862 \) (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

\(\Q(\sqrt{-1}) \), 3.3.169.1, \(\Q(\zeta_{11})^+\), 6.0.1827904.1, 10.0.219503494144.1, 15.15.432659002790862279847129.1

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 $30$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{6}$ $30$ R R $15^{2}$ $30$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ $15^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{3}$ $15^{2}$ $15^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{6}$ $30$

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
11Data not computed
$13$13.15.10.1$x^{15} + 79092 x^{6} - 228488 x^{3} + 80199288$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$
13.15.10.1$x^{15} + 79092 x^{6} - 228488 x^{3} + 80199288$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$