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Properties

Label 14.0.28229671831...8979.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,29^{13}\cdot 31^{7}$
Root discriminant $126.95$
Ramified primes $29, 31$
Class number $159152$ (GRH)
Class group $[2, 2, 14, 2842]$ (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![232170751, -233359527, 233636092, -87114552, 40437938, -13821758, 6613602, -656208, 511395, -16527, 16306, -220, 219, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 219*x^12 - 220*x^11 + 16306*x^10 - 16527*x^9 + 511395*x^8 - 656208*x^7 + 6613602*x^6 - 13821758*x^5 + 40437938*x^4 - 87114552*x^3 + 233636092*x^2 - 233359527*x + 232170751)
 
gp: K = bnfinit(x^14 - x^13 + 219*x^12 - 220*x^11 + 16306*x^10 - 16527*x^9 + 511395*x^8 - 656208*x^7 + 6613602*x^6 - 13821758*x^5 + 40437938*x^4 - 87114552*x^3 + 233636092*x^2 - 233359527*x + 232170751, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 219 x^{12} - 220 x^{11} + 16306 x^{10} - 16527 x^{9} + 511395 x^{8} - 656208 x^{7} + 6613602 x^{6} - 13821758 x^{5} + 40437938 x^{4} - 87114552 x^{3} + 233636092 x^{2} - 233359527 x + 232170751 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-282296718315876607143916888979=-\,29^{13}\cdot 31^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $126.95$
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:  $29, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(899=29\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{899}(1,·)$, $\chi_{899}(898,·)$, $\chi_{899}(805,·)$, $\chi_{899}(807,·)$, $\chi_{899}(745,·)$, $\chi_{899}(683,·)$, $\chi_{899}(557,·)$, $\chi_{899}(526,·)$, $\chi_{899}(373,·)$, $\chi_{899}(342,·)$, $\chi_{899}(216,·)$, $\chi_{899}(154,·)$, $\chi_{899}(92,·)$, $\chi_{899}(94,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{17} a^{11} - \frac{8}{17} a^{10} - \frac{5}{17} a^{9} - \frac{2}{17} a^{8} + \frac{6}{17} a^{7} + \frac{5}{17} a^{6} + \frac{3}{17} a^{5} - \frac{1}{17} a^{4} + \frac{7}{17} a^{3} - \frac{1}{17} a$, $\frac{1}{17} a^{12} - \frac{1}{17} a^{10} - \frac{8}{17} a^{9} + \frac{7}{17} a^{8} + \frac{2}{17} a^{7} - \frac{8}{17} a^{6} + \frac{6}{17} a^{5} - \frac{1}{17} a^{4} + \frac{5}{17} a^{3} - \frac{1}{17} a^{2} - \frac{8}{17} a$, $\frac{1}{45496369386705545285239680551015051375386704478217807} a^{13} - \frac{1091083610701963123770855689796284447102197537637514}{45496369386705545285239680551015051375386704478217807} a^{12} + \frac{1292859010205682503968589936327691852341628456262458}{45496369386705545285239680551015051375386704478217807} a^{11} - \frac{10202367240583960385390367016301035208951355528128839}{45496369386705545285239680551015051375386704478217807} a^{10} + \frac{18841793243146993351310444002385119085778285456569911}{45496369386705545285239680551015051375386704478217807} a^{9} - \frac{21520784427847083653183829552924208139859862838211499}{45496369386705545285239680551015051375386704478217807} a^{8} + \frac{20423935038058896434349762977192736670831979464691443}{45496369386705545285239680551015051375386704478217807} a^{7} + \frac{18967746049650865035072950976230488641672062530037336}{45496369386705545285239680551015051375386704478217807} a^{6} + \frac{16352621614596963165403772756678158524783513094762774}{45496369386705545285239680551015051375386704478217807} a^{5} + \frac{11372240597140755319661698979169755536796250264730962}{45496369386705545285239680551015051375386704478217807} a^{4} - \frac{14179396201697192778491895373897998970701706685053838}{45496369386705545285239680551015051375386704478217807} a^{3} - \frac{178759781012215122053148554719310731680917851477961}{771124904859416021783723399169746633481130584376573} a^{2} - \frac{921690054029915568653249096571431127563480036313139}{2676257022747385016778804738295003022081570851659871} a + \frac{47315154807208825915416388324885274872735121038933}{157426883691022648045812043429117824828327697156463}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{14}\times C_{2842}$, which has order $159152$ (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:  $6$
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:  \( 6020.985100147561 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{14}$ (as 14T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-899}) \), 7.7.594823321.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ R R ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{14}$

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
$29$29.14.13.11$x^{14} + 3712$$14$$1$$13$$C_{14}$$[\ ]_{14}$
$31$31.14.7.1$x^{14} - 59582 x^{8} + 887503681 x^{2} - 8914086971964$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$