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Properties

Label 28.0.13517157994...4944.3
Degree $28$
Signature $[0, 14]$
Discriminant $2^{28}\cdot 3^{14}\cdot 29^{26}$
Root discriminant $78.98$
Ramified primes $2, 3, 29$
Class number $1152$ (GRH)
Class group $[2, 4, 12, 12]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3728761, 0, -4216942, 0, -1962803, 0, 3918063, 0, 516968, 0, -596165, 0, 322699, 0, -10755, 0, -13066, 0, 8804, 0, -2160, 0, -243, 0, 170, 0, -23, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 23*x^26 + 170*x^24 - 243*x^22 - 2160*x^20 + 8804*x^18 - 13066*x^16 - 10755*x^14 + 322699*x^12 - 596165*x^10 + 516968*x^8 + 3918063*x^6 - 1962803*x^4 - 4216942*x^2 + 3728761)
 
gp: K = bnfinit(x^28 - 23*x^26 + 170*x^24 - 243*x^22 - 2160*x^20 + 8804*x^18 - 13066*x^16 - 10755*x^14 + 322699*x^12 - 596165*x^10 + 516968*x^8 + 3918063*x^6 - 1962803*x^4 - 4216942*x^2 + 3728761, 1)
 

Normalized defining polynomial

\( x^{28} - 23 x^{26} + 170 x^{24} - 243 x^{22} - 2160 x^{20} + 8804 x^{18} - 13066 x^{16} - 10755 x^{14} + 322699 x^{12} - 596165 x^{10} + 516968 x^{8} + 3918063 x^{6} - 1962803 x^{4} - 4216942 x^{2} + 3728761 \)

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

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 14]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(135171579942192030712001098144632895138136441638354944=2^{28}\cdot 3^{14}\cdot 29^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.98$
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:  $2, 3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(348=2^{2}\cdot 3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{348}(149,·)$, $\chi_{348}(1,·)$, $\chi_{348}(67,·)$, $\chi_{348}(227,·)$, $\chi_{348}(5,·)$, $\chi_{348}(151,·)$, $\chi_{348}(335,·)$, $\chi_{348}(245,·)$, $\chi_{348}(209,·)$, $\chi_{348}(83,·)$, $\chi_{348}(277,·)$, $\chi_{348}(23,·)$, $\chi_{348}(25,·)$, $\chi_{348}(283,·)$, $\chi_{348}(187,·)$, $\chi_{348}(295,·)$, $\chi_{348}(169,·)$, $\chi_{348}(107,·)$, $\chi_{348}(173,·)$, $\chi_{348}(239,·)$, $\chi_{348}(49,·)$, $\chi_{348}(91,·)$, $\chi_{348}(115,·)$, $\chi_{348}(181,·)$, $\chi_{348}(313,·)$, $\chi_{348}(59,·)$, $\chi_{348}(125,·)$, $\chi_{348}(341,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} - \frac{5}{17} a^{12} + \frac{8}{17} a^{10} - \frac{6}{17} a^{8} - \frac{4}{17} a^{6} + \frac{3}{17} a^{4} + \frac{2}{17} a^{2} + \frac{7}{17}$, $\frac{1}{17} a^{15} - \frac{5}{17} a^{13} + \frac{8}{17} a^{11} - \frac{6}{17} a^{9} - \frac{4}{17} a^{7} + \frac{3}{17} a^{5} + \frac{2}{17} a^{3} + \frac{7}{17} a$, $\frac{1}{17} a^{16} + \frac{1}{17}$, $\frac{1}{17} a^{17} + \frac{1}{17} a$, $\frac{1}{17} a^{18} + \frac{1}{17} a^{2}$, $\frac{1}{17} a^{19} + \frac{1}{17} a^{3}$, $\frac{1}{17} a^{20} + \frac{1}{17} a^{4}$, $\frac{1}{17} a^{21} + \frac{1}{17} a^{5}$, $\frac{1}{17} a^{22} + \frac{1}{17} a^{6}$, $\frac{1}{17} a^{23} + \frac{1}{17} a^{7}$, $\frac{1}{11849} a^{24} + \frac{62}{11849} a^{22} - \frac{161}{11849} a^{20} - \frac{144}{11849} a^{18} + \frac{80}{11849} a^{16} - \frac{312}{11849} a^{14} - \frac{3948}{11849} a^{12} + \frac{5086}{11849} a^{10} + \frac{581}{11849} a^{8} + \frac{1463}{11849} a^{6} + \frac{5074}{11849} a^{4} + \frac{3227}{11849} a^{2} - \frac{3855}{11849}$, $\frac{1}{11849} a^{25} + \frac{62}{11849} a^{23} - \frac{161}{11849} a^{21} - \frac{144}{11849} a^{19} + \frac{80}{11849} a^{17} - \frac{312}{11849} a^{15} - \frac{3948}{11849} a^{13} + \frac{5086}{11849} a^{11} + \frac{581}{11849} a^{9} + \frac{1463}{11849} a^{7} + \frac{5074}{11849} a^{5} + \frac{3227}{11849} a^{3} - \frac{3855}{11849} a$, $\frac{1}{1460221973464489184256054597610403932543} a^{26} - \frac{5529588300095160569722295175670422}{1460221973464489184256054597610403932543} a^{24} + \frac{2432374274383085773122151201269708054}{1460221973464489184256054597610403932543} a^{22} + \frac{9080289909388768016998968188406185285}{1460221973464489184256054597610403932543} a^{20} - \frac{30716677000830694114488107579540407058}{1460221973464489184256054597610403932543} a^{18} + \frac{18717677718709426689306190563170496804}{1460221973464489184256054597610403932543} a^{16} + \frac{4585493059984634265318074942424400318}{1460221973464489184256054597610403932543} a^{14} + \frac{181109791616211294184981396517714431017}{1460221973464489184256054597610403932543} a^{12} + \frac{560208454826597325047347273747072296373}{1460221973464489184256054597610403932543} a^{10} + \frac{40938051901674910661040933149930614282}{1460221973464489184256054597610403932543} a^{8} - \frac{13019149278448987656249923805372309538}{1460221973464489184256054597610403932543} a^{6} + \frac{87815925876609383209731203055352513646}{1460221973464489184256054597610403932543} a^{4} - \frac{154796367872558602637123027447960668640}{1460221973464489184256054597610403932543} a^{2} + \frac{86448603603552063916464017919642131341}{1460221973464489184256054597610403932543}$, $\frac{1}{2819688630759928614798441427985689993740533} a^{27} - \frac{107713691020316598413147300007344886140}{2819688630759928614798441427985689993740533} a^{25} + \frac{4147348051167338543893481703570948657092}{2819688630759928614798441427985689993740533} a^{23} - \frac{46641986321362097871303933091902404774972}{2819688630759928614798441427985689993740533} a^{21} - \frac{54525500746882141588607110861290655992051}{2819688630759928614798441427985689993740533} a^{19} + \frac{30226790175272042771408786921819196622272}{2819688630759928614798441427985689993740533} a^{17} - \frac{26259569052194922905428716180887341529929}{2819688630759928614798441427985689993740533} a^{15} + \frac{952553063545897961651796967941517597733704}{2819688630759928614798441427985689993740533} a^{13} - \frac{1378109803489703684059257279010665396462342}{2819688630759928614798441427985689993740533} a^{11} - \frac{585296970240387041442754077662144991568270}{2819688630759928614798441427985689993740533} a^{9} + \frac{938177688645474849315851489278345357298031}{2819688630759928614798441427985689993740533} a^{7} - \frac{389921957788863404960690575941070308916348}{2819688630759928614798441427985689993740533} a^{5} + \frac{1176828602398140687246042860341679591897945}{2819688630759928614798441427985689993740533} a^{3} + \frac{1263089109635468833785382960122725310735961}{2819688630759928614798441427985689993740533} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{12}\times C_{12}$, which has order $1152$ (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:  $13$
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:  \( 532329986474.6399 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{14}$ (as 28T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-29}) \), \(\Q(\sqrt{-87}) \), \(\Q(\sqrt{3}, \sqrt{-29})\), 7.7.594823321.1, 14.14.12677823379379991227056128.1, 14.0.168110140833113738264576.1, 14.0.22439994995240462987343.1

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 ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
$2$2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
$3$3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
29Data not computed