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Properties

Label 28.0.12984931501...0096.2
Degree $28$
Signature $[0, 14]$
Discriminant $2^{42}\cdot 43^{26}$
Root discriminant $92.97$
Ramified primes $2, 43$
Class number $985$ (GRH)
Class group $[985]$ (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![11474513, 20949672, -2705282, -15784328, 6900339, 33378830, 14087468, -35926296, -22322709, 13208998, 22992711, 4195314, -6512215, -2252476, 539117, 688988, 208293, -147466, -31809, 11954, -1323, 4108, 140, -1112, 172, 96, -23, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 2*x^27 - 23*x^26 + 96*x^25 + 172*x^24 - 1112*x^23 + 140*x^22 + 4108*x^21 - 1323*x^20 + 11954*x^19 - 31809*x^18 - 147466*x^17 + 208293*x^16 + 688988*x^15 + 539117*x^14 - 2252476*x^13 - 6512215*x^12 + 4195314*x^11 + 22992711*x^10 + 13208998*x^9 - 22322709*x^8 - 35926296*x^7 + 14087468*x^6 + 33378830*x^5 + 6900339*x^4 - 15784328*x^3 - 2705282*x^2 + 20949672*x + 11474513)
 
gp: K = bnfinit(x^28 - 2*x^27 - 23*x^26 + 96*x^25 + 172*x^24 - 1112*x^23 + 140*x^22 + 4108*x^21 - 1323*x^20 + 11954*x^19 - 31809*x^18 - 147466*x^17 + 208293*x^16 + 688988*x^15 + 539117*x^14 - 2252476*x^13 - 6512215*x^12 + 4195314*x^11 + 22992711*x^10 + 13208998*x^9 - 22322709*x^8 - 35926296*x^7 + 14087468*x^6 + 33378830*x^5 + 6900339*x^4 - 15784328*x^3 - 2705282*x^2 + 20949672*x + 11474513, 1)
 

Normalized defining polynomial

\( x^{28} - 2 x^{27} - 23 x^{26} + 96 x^{25} + 172 x^{24} - 1112 x^{23} + 140 x^{22} + 4108 x^{21} - 1323 x^{20} + 11954 x^{19} - 31809 x^{18} - 147466 x^{17} + 208293 x^{16} + 688988 x^{15} + 539117 x^{14} - 2252476 x^{13} - 6512215 x^{12} + 4195314 x^{11} + 22992711 x^{10} + 13208998 x^{9} - 22322709 x^{8} - 35926296 x^{7} + 14087468 x^{6} + 33378830 x^{5} + 6900339 x^{4} - 15784328 x^{3} - 2705282 x^{2} + 20949672 x + 11474513 \)

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:  \(12984931501152231903148118403201067499769297367881220096=2^{42}\cdot 43^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.97$
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:  $2, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(344=2^{3}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{344}(1,·)$, $\chi_{344}(213,·)$, $\chi_{344}(133,·)$, $\chi_{344}(193,·)$, $\chi_{344}(137,·)$, $\chi_{344}(269,·)$, $\chi_{344}(45,·)$, $\chi_{344}(145,·)$, $\chi_{344}(237,·)$, $\chi_{344}(21,·)$, $\chi_{344}(333,·)$, $\chi_{344}(217,·)$, $\chi_{344}(285,·)$, $\chi_{344}(161,·)$, $\chi_{344}(113,·)$, $\chi_{344}(293,·)$, $\chi_{344}(65,·)$, $\chi_{344}(97,·)$, $\chi_{344}(41,·)$, $\chi_{344}(257,·)$, $\chi_{344}(173,·)$, $\chi_{344}(125,·)$, $\chi_{344}(305,·)$, $\chi_{344}(309,·)$, $\chi_{344}(297,·)$, $\chi_{344}(121,·)$, $\chi_{344}(317,·)$, $\chi_{344}(85,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{79} a^{22} - \frac{7}{79} a^{21} - \frac{4}{79} a^{19} + \frac{12}{79} a^{18} - \frac{26}{79} a^{17} + \frac{27}{79} a^{16} + \frac{23}{79} a^{15} - \frac{2}{79} a^{14} + \frac{25}{79} a^{13} - \frac{26}{79} a^{12} - \frac{21}{79} a^{11} - \frac{9}{79} a^{10} - \frac{12}{79} a^{9} - \frac{29}{79} a^{8} - \frac{12}{79} a^{7} + \frac{3}{79} a^{6} + \frac{1}{79} a^{5} - \frac{23}{79} a^{4} - \frac{26}{79} a^{3} - \frac{20}{79} a^{2} + \frac{12}{79} a$, $\frac{1}{79} a^{23} + \frac{30}{79} a^{21} - \frac{4}{79} a^{20} - \frac{16}{79} a^{19} - \frac{21}{79} a^{18} + \frac{3}{79} a^{17} - \frac{25}{79} a^{16} + \frac{1}{79} a^{15} + \frac{11}{79} a^{14} - \frac{9}{79} a^{13} + \frac{34}{79} a^{12} + \frac{2}{79} a^{11} + \frac{4}{79} a^{10} - \frac{34}{79} a^{9} + \frac{22}{79} a^{8} - \frac{2}{79} a^{7} + \frac{22}{79} a^{6} - \frac{16}{79} a^{5} - \frac{29}{79} a^{4} + \frac{35}{79} a^{3} + \frac{30}{79} a^{2} + \frac{5}{79} a$, $\frac{1}{553} a^{24} + \frac{2}{553} a^{23} - \frac{2}{553} a^{22} - \frac{36}{553} a^{21} - \frac{103}{553} a^{20} + \frac{22}{79} a^{19} - \frac{186}{553} a^{18} + \frac{181}{553} a^{17} - \frac{44}{553} a^{16} - \frac{249}{553} a^{15} - \frac{239}{553} a^{14} + \frac{164}{553} a^{13} + \frac{16}{79} a^{12} + \frac{206}{553} a^{11} - \frac{54}{553} a^{10} + \frac{37}{79} a^{9} + \frac{37}{79} a^{8} + \frac{86}{553} a^{7} + \frac{90}{553} a^{6} + \frac{223}{553} a^{5} + \frac{2}{553} a^{4} - \frac{95}{553} a^{3} + \frac{73}{553} a^{2} + \frac{258}{553} a - \frac{3}{7}$, $\frac{1}{553} a^{25} + \frac{1}{553} a^{23} + \frac{3}{553} a^{22} - \frac{66}{553} a^{21} - \frac{221}{553} a^{20} - \frac{193}{553} a^{19} + \frac{39}{79} a^{18} - \frac{27}{79} a^{17} + \frac{8}{79} a^{16} - \frac{5}{79} a^{15} + \frac{96}{553} a^{14} + \frac{43}{553} a^{13} - \frac{137}{553} a^{12} - \frac{81}{553} a^{11} + \frac{80}{553} a^{10} + \frac{27}{79} a^{9} - \frac{187}{553} a^{8} + \frac{37}{553} a^{7} - \frac{251}{553} a^{6} + \frac{32}{553} a^{5} - \frac{1}{553} a^{4} + \frac{151}{553} a^{3} + \frac{25}{79} a^{2} + \frac{255}{553} a - \frac{1}{7}$, $\frac{1}{35685854565556697789} a^{26} - \frac{5397765606232999}{35685854565556697789} a^{25} - \frac{23551432453541178}{35685854565556697789} a^{24} + \frac{76582592718664670}{35685854565556697789} a^{23} + \frac{49407824492517578}{35685854565556697789} a^{22} - \frac{5113403783700852740}{35685854565556697789} a^{21} - \frac{9012483573040024438}{35685854565556697789} a^{20} - \frac{10017125306751952365}{35685854565556697789} a^{19} - \frac{15297037563647217617}{35685854565556697789} a^{18} + \frac{2365878917126591501}{35685854565556697789} a^{17} + \frac{5751167001255225049}{35685854565556697789} a^{16} + \frac{5746607569500485811}{35685854565556697789} a^{15} - \frac{11388178123702205038}{35685854565556697789} a^{14} + \frac{12084248424937286349}{35685854565556697789} a^{13} + \frac{7978721833373278056}{35685854565556697789} a^{12} + \frac{3453275778277399513}{35685854565556697789} a^{11} + \frac{4156139822531677}{15127534788281771} a^{10} + \frac{250025132693675102}{964482555825856697} a^{9} + \frac{11762748871987961404}{35685854565556697789} a^{8} - \frac{1605978039162743845}{5097979223650956827} a^{7} - \frac{13892188342745862706}{35685854565556697789} a^{6} + \frac{7519268356461043694}{35685854565556697789} a^{5} + \frac{296324362430622282}{5097979223650956827} a^{4} + \frac{15635071634281950763}{35685854565556697789} a^{3} + \frac{10410817530998934184}{35685854565556697789} a^{2} - \frac{14069184765946453151}{35685854565556697789} a - \frac{12864732471990}{191487782130149}$, $\frac{1}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{27} + \frac{118455504730714015068416605814638349328216746229535197828858248042434548}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{26} - \frac{2265907773604842662346019065546584026061176501238844940785140179818379474746816006585776}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{25} + \frac{2414316085187320129926151632051105891284933924734892606748495763725415663106106301567}{57250711854515832371463646149279529486260531373431796121746425977936389917152266491197909} a^{24} - \frac{32354824801175242967046501487918601989017846539336490132470690338323543656599296035115029}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{23} - \frac{353534663697636196600444283623013421397734255625653021853697516886509011679374898628412}{388376450688742538519929059012680051379767388506253535852928457309784699167708618629477707} a^{22} + \frac{6421744549982440752900576016255013574422350362833526752228709408707248415680310257945092808}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{21} + \frac{6821540231691067475925488982676879531730349612310253680778577990753307373193600421054810222}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{20} - \frac{1695815951768311114643761530693461638295103376753042077853272134300936104268942391691070475}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{19} - \frac{19140864753460205972466092548894509153355471273745663121068183286215210347403143147019969}{2052846953640496275033910740495594557293056196390197260936907560066004838457888412755810737} a^{18} + \frac{179265135683831433199764098649684666567956462920346739478007173417641140136369974293427378}{2052846953640496275033910740495594557293056196390197260936907560066004838457888412755810737} a^{17} + \frac{7059774213096135983549173903338876101371837593025017153896825717554594998625789861007868005}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{16} - \frac{4260827863000679085709086826013115863941856411605077524778428013534948744495586034575392535}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{15} + \frac{1130054769070077444519766221563175423681229589127882426939697873791819292010108289764223653}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{14} - \frac{2467584172475911247805554109458187910732041881533700919579485227712340952873444455111108074}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{13} + \frac{2888899035466226438773381620235322508326915400114836687867730194704240984586277892897290352}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{12} - \frac{3019647244725188422144954539313057125322133279947190176455256156787937198790971110482949446}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{11} + \frac{1424579509213464890434765282760695757752280420105509652752792260576843894577806262782868050}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{10} - \frac{80771559452524107605629613634059262228021018047992627200200232890262381762556324067941557}{2052846953640496275033910740495594557293056196390197260936907560066004838457888412755810737} a^{9} + \frac{5723388254397817986510045972909929223319543495638200530260181895038752536322043968603944738}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{8} - \frac{97856800481198195239100869093499622898754687387795494895642754010857579431381702748379705}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{7} + \frac{4585711334098803408906519597965299499607164005706043124609638933151658222752232982412217}{181897831335233847154903483335052682291789789553561782614662695195721947711458466953046521} a^{6} + \frac{3629666867226066235150720118997305282638157554821821889299989506363768509395262759368738689}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{5} - \frac{11694850246807596068456369911610095482739301165554805466621754090715905974069218230993547}{25985404476461978164986211905007526041684255650508826087808956456531706815922638136149503} a^{4} - \frac{6722248387106396194939475644331458578159753912526395282569318211966950153654795444395282931}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{3} - \frac{1904510658945148998251571212321518162108492879229447803500278910027846693927997841023514309}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a^{2} + \frac{3311851189558489483705044533481935624545352621530287190446976243227352405532894063492217909}{14369928675483473925237375183469161901051393374731380826558352920462033869205218889290675159} a - \frac{544174963901571848227298591228657147755815859468859106702235613851706231625645119582}{1252334515241167439980884172031454572499189584318862232023124024563136916504013624743}$

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

Class group and class number

$C_{985}$, which has order $985$ (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:  \( 9818745262125.305 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-43}) \), \(\Q(\sqrt{-86}) \), \(\Q(\sqrt{2}, \sqrt{-43})\), 7.7.6321363049.1, 14.14.83801419645740806624509952.1, 14.0.1718264124282290785243.1, 14.0.3603461044766854684853927936.1

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 ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{4}$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$

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
2Data not computed
43Data not computed