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Properties

Label 28.0.13517157994...4944.2
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 $2688$ (GRH)
Class group $[2, 4, 4, 84]$ (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![2825761, 0, 4370782, 0, 15557681, 0, -3636433, 0, 1311812, 0, -180077, 0, 165323, 0, -121607, 0, -4122, 0, 8564, 0, 2216, 0, -1731, 0, 354, 0, -31, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 31*x^26 + 354*x^24 - 1731*x^22 + 2216*x^20 + 8564*x^18 - 4122*x^16 - 121607*x^14 + 165323*x^12 - 180077*x^10 + 1311812*x^8 - 3636433*x^6 + 15557681*x^4 + 4370782*x^2 + 2825761)
 
gp: K = bnfinit(x^28 - 31*x^26 + 354*x^24 - 1731*x^22 + 2216*x^20 + 8564*x^18 - 4122*x^16 - 121607*x^14 + 165323*x^12 - 180077*x^10 + 1311812*x^8 - 3636433*x^6 + 15557681*x^4 + 4370782*x^2 + 2825761, 1)
 

Normalized defining polynomial

\( x^{28} - 31 x^{26} + 354 x^{24} - 1731 x^{22} + 2216 x^{20} + 8564 x^{18} - 4122 x^{16} - 121607 x^{14} + 165323 x^{12} - 180077 x^{10} + 1311812 x^{8} - 3636433 x^{6} + 15557681 x^{4} + 4370782 x^{2} + 2825761 \)

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$$\chi_{348}(149,·)$, $\chi_{348}(1,·)$, $\chi_{348}(323,·)$, $\chi_{348}(5,·)$, $\chi_{348}(7,·)$, $\chi_{348}(103,·)$, $\chi_{348}(139,·)$, $\chi_{348}(245,·)$, $\chi_{348}(209,·)$, $\chi_{348}(277,·)$, $\chi_{348}(343,·)$, $\chi_{348}(25,·)$, $\chi_{348}(71,·)$, $\chi_{348}(347,·)$, $\chi_{348}(199,·)$, $\chi_{348}(223,·)$, $\chi_{348}(35,·)$, $\chi_{348}(167,·)$, $\chi_{348}(169,·)$, $\chi_{348}(299,·)$, $\chi_{348}(173,·)$, $\chi_{348}(175,·)$, $\chi_{348}(49,·)$, $\chi_{348}(179,·)$, $\chi_{348}(181,·)$, $\chi_{348}(313,·)$, $\chi_{348}(125,·)$, $\chi_{348}(341,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17} a^{12} - \frac{4}{17} a^{8} - \frac{1}{17} a^{4} + \frac{4}{17}$, $\frac{1}{17} a^{13} - \frac{4}{17} a^{9} - \frac{1}{17} a^{5} + \frac{4}{17} a$, $\frac{1}{17} a^{14} - \frac{4}{17} a^{10} - \frac{1}{17} a^{6} + \frac{4}{17} a^{2}$, $\frac{1}{17} a^{15} - \frac{4}{17} a^{11} - \frac{1}{17} a^{7} + \frac{4}{17} a^{3}$, $\frac{1}{17} a^{16} - \frac{1}{17}$, $\frac{1}{17} a^{17} - \frac{1}{17} a$, $\frac{1}{17} a^{18} - \frac{1}{17} a^{2}$, $\frac{1}{17} a^{19} - \frac{1}{17} a^{3}$, $\frac{1}{289} a^{20} + \frac{3}{289} a^{18} + \frac{5}{289} a^{16} - \frac{7}{289} a^{14} - \frac{4}{289} a^{12} + \frac{79}{289} a^{10} + \frac{101}{289} a^{8} + \frac{143}{289} a^{6} - \frac{99}{289} a^{4} - \frac{133}{289} a^{2} - \frac{89}{289}$, $\frac{1}{289} a^{21} + \frac{3}{289} a^{19} + \frac{5}{289} a^{17} - \frac{7}{289} a^{15} - \frac{4}{289} a^{13} + \frac{79}{289} a^{11} + \frac{101}{289} a^{9} + \frac{143}{289} a^{7} - \frac{99}{289} a^{5} - \frac{133}{289} a^{3} - \frac{89}{289} a$, $\frac{1}{289} a^{22} - \frac{4}{289} a^{18} - \frac{5}{289} a^{16} + \frac{6}{289} a^{12} - \frac{4}{17} a^{10} - \frac{109}{289} a^{8} + \frac{67}{289} a^{6} - \frac{40}{289} a^{4} - \frac{47}{289} a^{2} - \frac{90}{289}$, $\frac{1}{11849} a^{23} + \frac{3}{11849} a^{21} - \frac{46}{11849} a^{19} + \frac{78}{11849} a^{17} + \frac{285}{11849} a^{15} - \frac{74}{11849} a^{13} - \frac{5101}{11849} a^{11} - \frac{2135}{11849} a^{9} + \frac{1924}{11849} a^{7} + \frac{3777}{11849} a^{5} - \frac{4662}{11849} a^{3} + \frac{44}{697} a$, $\frac{1}{93263479} a^{24} - \frac{133124}{93263479} a^{22} - \frac{115871}{93263479} a^{20} + \frac{1503138}{93263479} a^{18} - \frac{466623}{93263479} a^{16} - \frac{1002196}{93263479} a^{14} + \frac{2359615}{93263479} a^{12} - \frac{9827703}{93263479} a^{10} + \frac{8614179}{93263479} a^{8} - \frac{44994010}{93263479} a^{6} - \frac{18284553}{93263479} a^{4} + \frac{2405029}{93263479} a^{2} + \frac{1008146}{2274719}$, $\frac{1}{93263479} a^{25} + \frac{683}{93263479} a^{23} - \frac{37161}{93263479} a^{21} - \frac{134030}{93263479} a^{19} - \frac{2615406}{93263479} a^{17} + \frac{989167}{93263479} a^{15} - \frac{765172}{93263479} a^{13} - \frac{4680069}{93263479} a^{11} + \frac{41451991}{93263479} a^{9} + \frac{18178636}{93263479} a^{7} - \frac{46014086}{93263479} a^{5} + \frac{8528667}{93263479} a^{3} + \frac{16532465}{93263479} a$, $\frac{1}{2473498818311442779030586981632353} a^{26} + \frac{8899758832236754667088011}{2473498818311442779030586981632353} a^{24} - \frac{3546735758430857552249257274584}{2473498818311442779030586981632353} a^{22} + \frac{4241409262438906701403489632589}{2473498818311442779030586981632353} a^{20} + \frac{12176752784963283978519172285964}{2473498818311442779030586981632353} a^{18} - \frac{38537027374632604881728504173110}{2473498818311442779030586981632353} a^{16} + \frac{2933986485114133756574743883091}{2473498818311442779030586981632353} a^{14} - \frac{65372380445768911134117596662868}{2473498818311442779030586981632353} a^{12} - \frac{1175672041960556609154827022369479}{2473498818311442779030586981632353} a^{10} + \frac{581664341035846370936987896692870}{2473498818311442779030586981632353} a^{8} + \frac{962453530774272540546699726262820}{2473498818311442779030586981632353} a^{6} - \frac{189315206444526075840655182490895}{2473498818311442779030586981632353} a^{4} - \frac{517923876552293017169625919681721}{2473498818311442779030586981632353} a^{2} - \frac{11303240689649584719821850199788}{60329239471010799488550901991033}$, $\frac{1}{101413451550769153940254066246926473} a^{27} - \frac{521532698724154162088352129}{101413451550769153940254066246926473} a^{25} - \frac{4117772820613190193682326357301}{101413451550769153940254066246926473} a^{23} + \frac{40444193617726043536791574575792}{101413451550769153940254066246926473} a^{21} + \frac{2908724788912530687539266241408059}{101413451550769153940254066246926473} a^{19} + \frac{108565392611799406483386472600453}{101413451550769153940254066246926473} a^{17} - \frac{2738069027966947840232065223249138}{101413451550769153940254066246926473} a^{15} - \frac{2768021786492021216578563682400215}{101413451550769153940254066246926473} a^{13} + \frac{21765915700887752734083052457975860}{101413451550769153940254066246926473} a^{11} - \frac{7009256561657130495528245379281265}{101413451550769153940254066246926473} a^{9} + \frac{39926076820088135755548681887213454}{101413451550769153940254066246926473} a^{7} + \frac{5002456110301806168501034241042295}{101413451550769153940254066246926473} a^{5} + \frac{17371237421365976753753676639628498}{101413451550769153940254066246926473} a^{3} + \frac{17751081271344881451915457935023103}{101413451550769153940254066246926473} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{84}$, which has order $2688$ (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:  \( \frac{2729773008375457211}{17614789467148212613700969} a^{27} - \frac{84186284058644055375}{17614789467148212613700969} a^{25} + \frac{949112912447047986756}{17614789467148212613700969} a^{23} - \frac{4466117280551868347775}{17614789467148212613700969} a^{21} + \frac{4256287364530340094435}{17614789467148212613700969} a^{19} + \frac{28175401974806682070614}{17614789467148212613700969} a^{17} - \frac{6386052889700357177231}{17614789467148212613700969} a^{15} - \frac{369537423585515603088009}{17614789467148212613700969} a^{13} + \frac{382076733259297499664894}{17614789467148212613700969} a^{11} - \frac{259981068865718655103}{38044901656907586638663} a^{9} + \frac{3362786515297414681391643}{17614789467148212613700969} a^{7} - \frac{8483773797085126506553065}{17614789467148212613700969} a^{5} + \frac{37753029883558878099525217}{17614789467148212613700969} a^{3} + \frac{27180174763845314591565123}{17614789467148212613700969} a \) (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:  \( 4155697700585.085 \) (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

\(\Q(\sqrt{87}) \), \(\Q(\sqrt{-87}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{87})\), 7.7.594823321.1, 14.14.367656878002019745584627712.1, 14.0.22439994995240462987343.1, 14.0.5796901408038404767744.1

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 ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{28}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{28}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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$2.14.14.38$x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.38$x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
3Data not computed
$29$29.14.13.11$x^{14} + 3712$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.11$x^{14} + 3712$$14$$1$$13$$C_{14}$$[\ ]_{14}$