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Properties

Label 14.0.13760207184...6523.5
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 29^{13}$
Root discriminant $736.28$
Ramified primes $7, 29$
Class number $91756$ (GRH)
Class group $[91756]$ (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![699223478963, 606858815682, 323107507394, -72797482464, -12742935849, 1609330814, 302972831, 7780816, 366415, -73486, -22533, -812, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 812*x^11 - 22533*x^10 - 73486*x^9 + 366415*x^8 + 7780816*x^7 + 302972831*x^6 + 1609330814*x^5 - 12742935849*x^4 - 72797482464*x^3 + 323107507394*x^2 + 606858815682*x + 699223478963)
 
gp: K = bnfinit(x^14 - 812*x^11 - 22533*x^10 - 73486*x^9 + 366415*x^8 + 7780816*x^7 + 302972831*x^6 + 1609330814*x^5 - 12742935849*x^4 - 72797482464*x^3 + 323107507394*x^2 + 606858815682*x + 699223478963, 1)
 

Normalized defining polynomial

\( x^{14} - 812 x^{11} - 22533 x^{10} - 73486 x^{9} + 366415 x^{8} + 7780816 x^{7} + 302972831 x^{6} + 1609330814 x^{5} - 12742935849 x^{4} - 72797482464 x^{3} + 323107507394 x^{2} + 606858815682 x + 699223478963 \)

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:  \(-13760207184971837367389621368916758066523=-\,7^{25}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $736.28$
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:  $7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1421=7^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1421}(1,·)$, $\chi_{1421}(643,·)$, $\chi_{1421}(778,·)$, $\chi_{1421}(1420,·)$, $\chi_{1421}(78,·)$, $\chi_{1421}(1359,·)$, $\chi_{1421}(848,·)$, $\chi_{1421}(1002,·)$, $\chi_{1421}(400,·)$, $\chi_{1421}(419,·)$, $\chi_{1421}(573,·)$, $\chi_{1421}(1021,·)$, $\chi_{1421}(62,·)$, $\chi_{1421}(1343,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{16} a^{2} + \frac{1}{16} a + \frac{1}{16}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{3}{16} a^{3} - \frac{1}{8} a^{2} - \frac{1}{8} a - \frac{1}{16}$, $\frac{1}{64} a^{10} - \frac{1}{32} a^{9} + \frac{1}{64} a^{8} - \frac{1}{32} a^{7} - \frac{1}{64} a^{6} - \frac{3}{32} a^{5} - \frac{7}{64} a^{4} + \frac{3}{16} a^{3} + \frac{25}{64} a^{2} + \frac{7}{16} a + \frac{11}{64}$, $\frac{1}{64} a^{11} + \frac{1}{64} a^{9} + \frac{3}{64} a^{7} + \frac{1}{16} a^{6} - \frac{3}{64} a^{5} + \frac{7}{32} a^{4} - \frac{3}{64} a^{3} + \frac{15}{32} a^{2} + \frac{11}{64} a + \frac{13}{32}$, $\frac{1}{512} a^{12} + \frac{3}{512} a^{11} - \frac{7}{512} a^{9} + \frac{3}{256} a^{8} + \frac{3}{512} a^{7} + \frac{1}{256} a^{6} + \frac{59}{512} a^{5} - \frac{49}{256} a^{4} - \frac{123}{512} a^{3} + \frac{7}{64} a^{2} - \frac{37}{512} a + \frac{251}{512}$, $\frac{1}{7488419203606572977001823651625543822960115058426474287594738176} a^{13} - \frac{1152817883198195590938065623772392497291442020060730275724647}{7488419203606572977001823651625543822960115058426474287594738176} a^{12} - \frac{4366116335228118777774308027285901147477055198181153871639231}{3744209601803286488500911825812771911480057529213237143797369088} a^{11} - \frac{45911071401800820683378101566917070863610479749925369192493535}{7488419203606572977001823651625543822960115058426474287594738176} a^{10} + \frac{36369360102148735050431106146768473956084667771892665688162615}{1872104800901643244250455912906385955740028764606618571898684544} a^{9} - \frac{70991865742366128107105016059239714407673657838943300868674065}{7488419203606572977001823651625543822960115058426474287594738176} a^{8} - \frac{90335558110231674955218940275553057814704542483292938615472579}{1872104800901643244250455912906385955740028764606618571898684544} a^{7} - \frac{809646357954440217966453729773992323871671046113574637126322049}{7488419203606572977001823651625543822960115058426474287594738176} a^{6} + \frac{1274591880695523089773886261873875119593449490171730095152463}{14625818757044087845706686819581140279218974723489207592958473} a^{5} - \frac{297864046119194366925063535568188350991091827677807656784479871}{7488419203606572977001823651625543822960115058426474287594738176} a^{4} + \frac{294472934428905955912708124103742411945996472383955283142430755}{3744209601803286488500911825812771911480057529213237143797369088} a^{3} - \frac{825871119611344462873779811233350980730374318307774456749958701}{7488419203606572977001823651625543822960115058426474287594738176} a^{2} + \frac{959068942879196517743028245180158998383041675071501900736677037}{7488419203606572977001823651625543822960115058426474287594738176} a - \frac{14427220627873726935473542401577713265381054627491795061750807}{61380485275463712926244456160865113302951762773987494160612608}$

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

Class group and class number

$C_{91756}$, which has order $91756$ (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:  \( 8981416884.6774 \) (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

\(\Q(\sqrt{-203}) \), 7.7.8233120419813614521.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$7$7.14.25.73$x^{14} + 35 x^{13} + 21 x^{12} + 98 x^{11} - 49 x^{10} + 147 x^{9} - 147 x^{8} + 14 x^{7} - 49 x^{6} + 147 x^{5} + 98 x^{4} + 147 x^{3} + 49 x^{2} - 49 x + 28$$14$$1$$25$$C_{14}$$[2]_{2}$
$29$29.14.13.2$x^{14} - 116$$14$$1$$13$$C_{14}$$[\ ]_{14}$