Properties

Label 28.0.503...649.1
Degree $28$
Signature $[0, 14]$
Discriminant $5.036\times 10^{44}$
Root discriminant $39.49$
Ramified primes $3, 29$
Class number $192$ (GRH)
Class group $[4, 4, 12]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 14*x^26 - 11*x^25 + 115*x^24 - 79*x^23 + 617*x^22 - 353*x^21 + 2421*x^20 - 1188*x^19 + 7015*x^18 - 2803*x^17 + 15415*x^16 - 5107*x^15 + 25195*x^14 - 6334*x^13 + 30532*x^12 - 6088*x^11 + 26057*x^10 - 3239*x^9 + 15207*x^8 - 1590*x^7 + 5362*x^6 - 70*x^5 + 1050*x^4 - 84*x^3 + 77*x^2 + 7*x + 1)
 
gp: K = bnfinit(x^28 - x^27 + 14*x^26 - 11*x^25 + 115*x^24 - 79*x^23 + 617*x^22 - 353*x^21 + 2421*x^20 - 1188*x^19 + 7015*x^18 - 2803*x^17 + 15415*x^16 - 5107*x^15 + 25195*x^14 - 6334*x^13 + 30532*x^12 - 6088*x^11 + 26057*x^10 - 3239*x^9 + 15207*x^8 - 1590*x^7 + 5362*x^6 - 70*x^5 + 1050*x^4 - 84*x^3 + 77*x^2 + 7*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, 77, -84, 1050, -70, 5362, -1590, 15207, -3239, 26057, -6088, 30532, -6334, 25195, -5107, 15415, -2803, 7015, -1188, 2421, -353, 617, -79, 115, -11, 14, -1, 1]);
 

\( x^{28} - x^{27} + 14 x^{26} - 11 x^{25} + 115 x^{24} - 79 x^{23} + 617 x^{22} - 353 x^{21} + 2421 x^{20} - 1188 x^{19} + 7015 x^{18} - 2803 x^{17} + 15415 x^{16} - 5107 x^{15} + 25195 x^{14} - 6334 x^{13} + 30532 x^{12} - 6088 x^{11} + 26057 x^{10} - 3239 x^{9} + 15207 x^{8} - 1590 x^{7} + 5362 x^{6} - 70 x^{5} + 1050 x^{4} - 84 x^{3} + 77 x^{2} + 7 x + 1 \)

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(503553375386417026489977159144851919778199649\)\(\medspace = 3^{14}\cdot 29^{26}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $39.49$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 29$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $28$
This field is Galois and abelian over $\Q$.
Conductor:  \(87=3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{87}(64,·)$, $\chi_{87}(1,·)$, $\chi_{87}(86,·)$, $\chi_{87}(67,·)$, $\chi_{87}(4,·)$, $\chi_{87}(5,·)$, $\chi_{87}(7,·)$, $\chi_{87}(74,·)$, $\chi_{87}(13,·)$, $\chi_{87}(16,·)$, $\chi_{87}(82,·)$, $\chi_{87}(83,·)$, $\chi_{87}(20,·)$, $\chi_{87}(22,·)$, $\chi_{87}(23,·)$, $\chi_{87}(25,·)$, $\chi_{87}(28,·)$, $\chi_{87}(80,·)$, $\chi_{87}(34,·)$, $\chi_{87}(35,·)$, $\chi_{87}(38,·)$, $\chi_{87}(71,·)$, $\chi_{87}(49,·)$, $\chi_{87}(52,·)$, $\chi_{87}(53,·)$, $\chi_{87}(59,·)$, $\chi_{87}(62,·)$, $\chi_{87}(65,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{17} a^{26} + \frac{6}{17} a^{25} + \frac{8}{17} a^{24} - \frac{5}{17} a^{23} + \frac{2}{17} a^{22} + \frac{5}{17} a^{21} - \frac{5}{17} a^{20} + \frac{1}{17} a^{19} - \frac{1}{17} a^{18} - \frac{2}{17} a^{17} - \frac{6}{17} a^{16} + \frac{5}{17} a^{15} - \frac{4}{17} a^{14} - \frac{3}{17} a^{13} + \frac{2}{17} a^{12} - \frac{5}{17} a^{11} + \frac{5}{17} a^{10} + \frac{1}{17} a^{9} + \frac{1}{17} a^{8} + \frac{1}{17} a^{7} + \frac{2}{17} a^{6} + \frac{8}{17} a^{5} + \frac{1}{17} a^{4} - \frac{5}{17} a^{3} - \frac{2}{17} a^{2} + \frac{6}{17} a - \frac{6}{17}$, $\frac{1}{3895907347685506947339847625632831500841} a^{27} + \frac{94974196297919567674530297273816680935}{3895907347685506947339847625632831500841} a^{26} - \frac{1757113464741665998113530038316256390903}{3895907347685506947339847625632831500841} a^{25} + \frac{24578954983254063362995042355463027681}{3895907347685506947339847625632831500841} a^{24} - \frac{307144181563506270526384407919579491408}{3895907347685506947339847625632831500841} a^{23} + \frac{424237376936331706001988848616728841772}{3895907347685506947339847625632831500841} a^{22} - \frac{1088401692781031861288783548562989117133}{3895907347685506947339847625632831500841} a^{21} - \frac{1802550801714419401257485572078452070133}{3895907347685506947339847625632831500841} a^{20} + \frac{687727648764306980147201738082910313485}{3895907347685506947339847625632831500841} a^{19} - \frac{1535578479753183437920119565859459594718}{3895907347685506947339847625632831500841} a^{18} + \frac{155154103626356511406811380802035561606}{3895907347685506947339847625632831500841} a^{17} - \frac{1778103150074836056109172676602648103048}{3895907347685506947339847625632831500841} a^{16} - \frac{1538325354226302230179671647617129862406}{3895907347685506947339847625632831500841} a^{15} - \frac{1818183532261965231037310205997831490806}{3895907347685506947339847625632831500841} a^{14} - \frac{1572084019506622842671319022828475296847}{3895907347685506947339847625632831500841} a^{13} - \frac{516485973011675242874436686084477689569}{3895907347685506947339847625632831500841} a^{12} + \frac{1725074057080875135854069638030427506029}{3895907347685506947339847625632831500841} a^{11} - \frac{36375549898676541406401752818992323114}{3895907347685506947339847625632831500841} a^{10} + \frac{1406999444225789946382404844468854014208}{3895907347685506947339847625632831500841} a^{9} + \frac{752043549368866031985994752216068354425}{3895907347685506947339847625632831500841} a^{8} - \frac{296748472292101331385986882560244289680}{3895907347685506947339847625632831500841} a^{7} - \frac{83384469760869518667619088021452574201}{229171020452088643961167507390166558873} a^{6} + \frac{810715063244245183211705340945781853939}{3895907347685506947339847625632831500841} a^{5} + \frac{250897107727719820246953299519325099740}{3895907347685506947339847625632831500841} a^{4} + \frac{912927810758554317674830797681254603670}{3895907347685506947339847625632831500841} a^{3} - \frac{989103005516448858452844619781883190451}{3895907347685506947339847625632831500841} a^{2} + \frac{1427458978851209173918301073010047240985}{3895907347685506947339847625632831500841} a - \frac{687849409610446700814666501434686311070}{3895907347685506947339847625632831500841}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{12}$, which has order $192$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{459718160048198964470148224851155793007}{3895907347685506947339847625632831500841} a^{27} + \frac{491963445984988705273217591819646684565}{3895907347685506947339847625632831500841} a^{26} - \frac{6448629807164339806286652834135729651835}{3895907347685506947339847625632831500841} a^{25} + \frac{5489048472554384718748952494620363866639}{3895907347685506947339847625632831500841} a^{24} - \frac{52951072199703845948981282554345438827087}{3895907347685506947339847625632831500841} a^{23} + \frac{39819307479811215431616417681920535084871}{3895907347685506947339847625632831500841} a^{22} - \frac{283987576946209170566480179195940845846935}{3895907347685506947339847625632831500841} a^{21} + \frac{180713865062290080350196112311444834870074}{3895907347685506947339847625632831500841} a^{20} - \frac{1112673865127612497690665577819613726783806}{3895907347685506947339847625632831500841} a^{19} + \frac{36344069922386707864807676303648846199636}{229171020452088643961167507390166558873} a^{18} - \frac{3217951691774023788210898570050862866706706}{3895907347685506947339847625632831500841} a^{17} + \frac{87876390160312095317718859986505853860340}{229171020452088643961167507390166558873} a^{16} - \frac{7047879163309816292391973949049811453306036}{3895907347685506947339847625632831500841} a^{15} + \frac{2797655820669082469076957101458329888476089}{3895907347685506947339847625632831500841} a^{14} - \frac{11468990009854706273972141903057069947674868}{3895907347685506947339847625632831500841} a^{13} + \frac{3642273987431523561845329655519556538625641}{3895907347685506947339847625632831500841} a^{12} - \frac{13797351382373102873603284632167040793349857}{3895907347685506947339847625632831500841} a^{11} + \frac{3691710695278352959379085590040921104499799}{3895907347685506947339847625632831500841} a^{10} - \frac{11655967246241058805749943256068328625815037}{3895907347685506947339847625632831500841} a^{9} + \frac{2250968703208521239456502251110489534766072}{3895907347685506947339847625632831500841} a^{8} - \frac{392647765542212833797689295166405365735365}{229171020452088643961167507390166558873} a^{7} + \frac{1198090561014405273787508711780039774147370}{3895907347685506947339847625632831500841} a^{6} - \frac{2288207023706649952564153692852833900212139}{3895907347685506947339847625632831500841} a^{5} + \frac{197340796559132081521154403951415280347726}{3895907347685506947339847625632831500841} a^{4} - \frac{415651585504784233687805415050104789263488}{3895907347685506947339847625632831500841} a^{3} + \frac{85650487097946204886580171571688541211264}{3895907347685506947339847625632831500841} a^{2} - \frac{27260512845676607370643210950855091389682}{3895907347685506947339847625632831500841} a + \frac{1495905399110794953147810115657088139776}{3895907347685506947339847625632831500841} \) (order $6$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 487075979.1876791 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 487075979.1876791 \cdot 192}{6\sqrt{503553375386417026489977159144851919778199649}}\approx 0.103810693395349$ (assuming GRH)

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{-87}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{29})\), 7.7.594823321.1, \(\Q(\zeta_{29})^+\), 14.0.22439994995240462987343.1, 14.0.773792930870360792667.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
29Data not computed