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Properties

Label 28.0.79097329061...8896.1
Degree $28$
Signature $[0, 14]$
Discriminant $2^{42}\cdot 7^{50}$
Root discriminant $91.34$
Ramified primes $2, 7$
Class number $1234$ (GRH)
Class group $[1234]$ (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![4391639, 999110, -17816547, -13710102, 62510966, -38195010, -11198019, 24278114, 5028527, -36011178, 11480798, 16406964, -8273370, -141078, 1760256, -1262142, -154847, 281638, 19348, -13468, 2345, -5444, -1862, 924, 378, -56, -28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 28*x^26 - 56*x^25 + 378*x^24 + 924*x^23 - 1862*x^22 - 5444*x^21 + 2345*x^20 - 13468*x^19 + 19348*x^18 + 281638*x^17 - 154847*x^16 - 1262142*x^15 + 1760256*x^14 - 141078*x^13 - 8273370*x^12 + 16406964*x^11 + 11480798*x^10 - 36011178*x^9 + 5028527*x^8 + 24278114*x^7 - 11198019*x^6 - 38195010*x^5 + 62510966*x^4 - 13710102*x^3 - 17816547*x^2 + 999110*x + 4391639)
 
gp: K = bnfinit(x^28 - 28*x^26 - 56*x^25 + 378*x^24 + 924*x^23 - 1862*x^22 - 5444*x^21 + 2345*x^20 - 13468*x^19 + 19348*x^18 + 281638*x^17 - 154847*x^16 - 1262142*x^15 + 1760256*x^14 - 141078*x^13 - 8273370*x^12 + 16406964*x^11 + 11480798*x^10 - 36011178*x^9 + 5028527*x^8 + 24278114*x^7 - 11198019*x^6 - 38195010*x^5 + 62510966*x^4 - 13710102*x^3 - 17816547*x^2 + 999110*x + 4391639, 1)
 

Normalized defining polynomial

\( x^{28} - 28 x^{26} - 56 x^{25} + 378 x^{24} + 924 x^{23} - 1862 x^{22} - 5444 x^{21} + 2345 x^{20} - 13468 x^{19} + 19348 x^{18} + 281638 x^{17} - 154847 x^{16} - 1262142 x^{15} + 1760256 x^{14} - 141078 x^{13} - 8273370 x^{12} + 16406964 x^{11} + 11480798 x^{10} - 36011178 x^{9} + 5028527 x^{8} + 24278114 x^{7} - 11198019 x^{6} - 38195010 x^{5} + 62510966 x^{4} - 13710102 x^{3} - 17816547 x^{2} + 999110 x + 4391639 \)

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:  \(7909732906157957559057675256932747228387119659164368896=2^{42}\cdot 7^{50}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.34$
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:  $2, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(392=2^{3}\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{392}(1,·)$, $\chi_{392}(197,·)$, $\chi_{392}(321,·)$, $\chi_{392}(265,·)$, $\chi_{392}(13,·)$, $\chi_{392}(337,·)$, $\chi_{392}(141,·)$, $\chi_{392}(209,·)$, $\chi_{392}(237,·)$, $\chi_{392}(85,·)$, $\chi_{392}(281,·)$, $\chi_{392}(153,·)$, $\chi_{392}(29,·)$, $\chi_{392}(69,·)$, $\chi_{392}(225,·)$, $\chi_{392}(293,·)$, $\chi_{392}(97,·)$, $\chi_{392}(41,·)$, $\chi_{392}(365,·)$, $\chi_{392}(349,·)$, $\chi_{392}(113,·)$, $\chi_{392}(253,·)$, $\chi_{392}(181,·)$, $\chi_{392}(169,·)$, $\chi_{392}(57,·)$, $\chi_{392}(377,·)$, $\chi_{392}(125,·)$, $\chi_{392}(309,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{79} a^{25} - \frac{21}{79} a^{24} + \frac{10}{79} a^{23} + \frac{34}{79} a^{22} - \frac{28}{79} a^{21} + \frac{32}{79} a^{20} - \frac{34}{79} a^{19} + \frac{8}{79} a^{18} + \frac{37}{79} a^{17} + \frac{5}{79} a^{16} + \frac{16}{79} a^{15} + \frac{8}{79} a^{14} - \frac{38}{79} a^{13} - \frac{36}{79} a^{12} - \frac{39}{79} a^{11} - \frac{22}{79} a^{10} + \frac{35}{79} a^{9} - \frac{12}{79} a^{8} + \frac{35}{79} a^{7} - \frac{24}{79} a^{6} + \frac{2}{79} a^{5} + \frac{22}{79} a^{4} - \frac{18}{79} a^{3} - \frac{19}{79} a^{2} - \frac{25}{79} a - \frac{8}{79}$, $\frac{1}{4018632481368851789} a^{26} + \frac{6478952677844414}{4018632481368851789} a^{25} - \frac{115783692989069473}{4018632481368851789} a^{24} + \frac{244652788859669721}{4018632481368851789} a^{23} - \frac{1813335070774494153}{4018632481368851789} a^{22} - \frac{989800009542384576}{4018632481368851789} a^{21} + \frac{1681617593294137666}{4018632481368851789} a^{20} - \frac{639304876382534036}{4018632481368851789} a^{19} - \frac{1262964557235728217}{4018632481368851789} a^{18} + \frac{258282010181015481}{4018632481368851789} a^{17} + \frac{1500849214044030609}{4018632481368851789} a^{16} - \frac{307027243745483763}{4018632481368851789} a^{15} + \frac{1787024403907720495}{4018632481368851789} a^{14} + \frac{94418405988095999}{4018632481368851789} a^{13} - \frac{1718358288709236117}{4018632481368851789} a^{12} - \frac{1913514260327475094}{4018632481368851789} a^{11} - \frac{841794148223614034}{4018632481368851789} a^{10} - \frac{9423498193817535}{59979589274161967} a^{9} - \frac{1905287859135586273}{4018632481368851789} a^{8} + \frac{1730730448558450788}{4018632481368851789} a^{7} - \frac{977314008986693283}{4018632481368851789} a^{6} + \frac{973777260362670106}{4018632481368851789} a^{5} + \frac{1380474469853392167}{4018632481368851789} a^{4} - \frac{13390793077149921}{4018632481368851789} a^{3} - \frac{1761635553255500862}{4018632481368851789} a^{2} - \frac{578441447542658276}{4018632481368851789} a + \frac{278851469964607743}{4018632481368851789}$, $\frac{1}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{27} + \frac{81416627306187379222860564203549707327049591324504599851017129852714378717181882}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{26} - \frac{92980883480151970262775895781248541866349432436441644097403096182293067605328867072338771761136456}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{25} - \frac{5470859170582322685381568301740290143411882770959909289952932034852514328410731745295072901648011976}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{24} + \frac{390883792738998301839353344592176901712571241509644871708812437683045104935446411621293468574006879}{818539005784192497938409890191952815749936270344053019174999168996864447178543497761089630361594471} a^{23} + \frac{2040228649718583371619375660402447122863207242845853879815411994091543626521211103013045825170870237}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{22} - \frac{1505161701492899055971274837753687395138081319150632245592461816626586080296242619162422596961295297}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{21} + \frac{4969045220756258503099331109687510581910190388225028680177871595997913397266846522330596764296430560}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{20} + \frac{6898510279286762116970499206238081063349608491223946240500450726777046928051202780002820534966287335}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{19} + \frac{3592938925117631535501025394683739942708461179253159268862998696051216809502532853284422331658612673}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{18} + \frac{7196422028424084832339235976876045077740305187430873496198307788141786874861842333342791812681109295}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{17} + \frac{379907306199027451632055696353556652339711449165706338382493259230621624750648188992405506599176454}{818539005784192497938409890191952815749936270344053019174999168996864447178543497761089630361594471} a^{16} - \frac{4780612908584551387562649826115165726793984306600204803634344988604051495039971911020043154871113071}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{15} + \frac{18969401280506665422384872816039931135393022263865346525431229649723337597011556028528627072749192}{196863811517717183048478328020849411382896065019455789421835243176461069574586410853932949074307531} a^{14} - \frac{861357318634977504840025949137623144989781486057481361958820142235480363889035018205519229939625777}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{13} + \frac{1777683709842351508372323009197556830641510267478676545493708801743221343399514635671347561737512160}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{12} - \frac{3190418226739487440277812004371885023211925281584955968499498486869547682927631221747959484579682329}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{11} - \frac{3279651979042192859510317178193601930504077372521452059774756264782544417709301906096912410923212032}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{10} - \frac{5162476209483851261026631415189090918260655874477419835381253429905202595112136930145248966391089142}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{9} - \frac{3110937149623382047144672405375083572175599396565354052248461727113437302165629796433085434774601746}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{8} + \frac{18089525042468743075837013895071275424777232709400582864549842544137408873509359067357396462383146}{78945386344668311983907552861152809640856797647395976468654742187514845159351910951577172471422817} a^{7} + \frac{7365838591436713367324646753558011563401783626992522668395399675896455821745716172878707944208095991}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{6} + \frac{4057561654778012007461340010335600542742893306459051011064248697335470940299302952433547688361501832}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{5} - \frac{768029634173423046417718658059562248861425074135839080847121672283226927242263384071668056172785870}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{4} + \frac{1757036528597650292713191888669902397373771196929593841615391702994321546264158741646277998286994735}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{3} - \frac{6316816883814535322440956814702145306474059691234044632772640375332238764960757472735136060931634710}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a^{2} + \frac{2310792419184339360045839479783325954467626447181455629257703703397021581916661917783746729615735208}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949} a - \frac{4877233129527957514176589162548334218065321279826562551750618423679877916729146270624113095978776194}{15552241109899657460829787913647103499248789136537007364324984210940424496392326457460702976870294949}$

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

Class group and class number

$C_{1234}$, which has order $1234$ (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:  \( 6078393112068.115 \) (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

\(\Q(\sqrt{-14}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{-7})\), 7.7.13841287201.1, 14.0.2812424737865523319657201664.1, 14.0.1341068619663964900807.1, 14.14.401774962552217617093885952.1

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 ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$

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
$2$2.14.21.34$x^{14} + 4 x^{13} + 8 x^{12} + 4 x^{11} + 5 x^{10} + 8 x^{9} - 6 x^{8} - 6 x^{7} + x^{6} + 6 x^{5} + 2 x^{3} + 7 x^{2} + 6 x - 7$$2$$7$$21$$C_{14}$$[3]^{7}$
2.14.21.34$x^{14} + 4 x^{13} + 8 x^{12} + 4 x^{11} + 5 x^{10} + 8 x^{9} - 6 x^{8} - 6 x^{7} + x^{6} + 6 x^{5} + 2 x^{3} + 7 x^{2} + 6 x - 7$$2$$7$$21$$C_{14}$$[3]^{7}$
$7$7.14.25.59$x^{14} - 168 x^{13} + 70 x^{12} - 147 x^{11} + 147 x^{10} - 98 x^{9} + 49 x^{8} + 168 x^{7} - 49 x^{4} - 147 x^{3} - 49 x^{2} + 98 x + 126$$14$$1$$25$$C_{14}$$[2]_{2}$
7.14.25.59$x^{14} - 168 x^{13} + 70 x^{12} - 147 x^{11} + 147 x^{10} - 98 x^{9} + 49 x^{8} + 168 x^{7} - 49 x^{4} - 147 x^{3} - 49 x^{2} + 98 x + 126$$14$$1$$25$$C_{14}$$[2]_{2}$