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Properties

Label 28.28.4145289608...6369.1
Degree $28$
Signature $[28, 0]$
Discriminant $13^{14}\cdot 29^{26}$
Root discriminant $82.21$
Ramified primes $13, 29$
Class number $1$ (GRH)
Class group Trivial (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![-233, -26357, 334177, -1286775, 51715, 12053439, -28078033, 1350369, 74956958, -79028331, -47279125, 117667962, -18213246, -71317048, 32941703, 20897646, -15762270, -2566528, 3851933, -118317, -531130, 77013, 40314, -9718, -1373, 548, -3, -12, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 12*x^27 - 3*x^26 + 548*x^25 - 1373*x^24 - 9718*x^23 + 40314*x^22 + 77013*x^21 - 531130*x^20 - 118317*x^19 + 3851933*x^18 - 2566528*x^17 - 15762270*x^16 + 20897646*x^15 + 32941703*x^14 - 71317048*x^13 - 18213246*x^12 + 117667962*x^11 - 47279125*x^10 - 79028331*x^9 + 74956958*x^8 + 1350369*x^7 - 28078033*x^6 + 12053439*x^5 + 51715*x^4 - 1286775*x^3 + 334177*x^2 - 26357*x - 233)
 
gp: K = bnfinit(x^28 - 12*x^27 - 3*x^26 + 548*x^25 - 1373*x^24 - 9718*x^23 + 40314*x^22 + 77013*x^21 - 531130*x^20 - 118317*x^19 + 3851933*x^18 - 2566528*x^17 - 15762270*x^16 + 20897646*x^15 + 32941703*x^14 - 71317048*x^13 - 18213246*x^12 + 117667962*x^11 - 47279125*x^10 - 79028331*x^9 + 74956958*x^8 + 1350369*x^7 - 28078033*x^6 + 12053439*x^5 + 51715*x^4 - 1286775*x^3 + 334177*x^2 - 26357*x - 233, 1)
 

Normalized defining polynomial

\( x^{28} - 12 x^{27} - 3 x^{26} + 548 x^{25} - 1373 x^{24} - 9718 x^{23} + 40314 x^{22} + 77013 x^{21} - 531130 x^{20} - 118317 x^{19} + 3851933 x^{18} - 2566528 x^{17} - 15762270 x^{16} + 20897646 x^{15} + 32941703 x^{14} - 71317048 x^{13} - 18213246 x^{12} + 117667962 x^{11} - 47279125 x^{10} - 79028331 x^{9} + 74956958 x^{8} + 1350369 x^{7} - 28078033 x^{6} + 12053439 x^{5} + 51715 x^{4} - 1286775 x^{3} + 334177 x^{2} - 26357 x - 233 \)

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

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[28, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(414528960816105641952550326302235364874378673563986369=13^{14}\cdot 29^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.21$
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:  $13, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(377=13\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{377}(64,·)$, $\chi_{377}(1,·)$, $\chi_{377}(194,·)$, $\chi_{377}(324,·)$, $\chi_{377}(326,·)$, $\chi_{377}(129,·)$, $\chi_{377}(207,·)$, $\chi_{377}(144,·)$, $\chi_{377}(209,·)$, $\chi_{377}(274,·)$, $\chi_{377}(339,·)$, $\chi_{377}(196,·)$, $\chi_{377}(25,·)$, $\chi_{377}(248,·)$, $\chi_{377}(92,·)$, $\chi_{377}(285,·)$, $\chi_{377}(352,·)$, $\chi_{377}(38,·)$, $\chi_{377}(103,·)$, $\chi_{377}(168,·)$, $\chi_{377}(233,·)$, $\chi_{377}(170,·)$, $\chi_{377}(51,·)$, $\chi_{377}(53,·)$, $\chi_{377}(183,·)$, $\chi_{377}(376,·)$, $\chi_{377}(313,·)$, $\chi_{377}(181,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{17} a^{26} + \frac{1}{17} a^{25} - \frac{4}{17} a^{24} + \frac{6}{17} a^{23} + \frac{2}{17} a^{22} - \frac{1}{17} a^{21} + \frac{1}{17} a^{18} - \frac{1}{17} a^{17} - \frac{5}{17} a^{16} - \frac{4}{17} a^{15} - \frac{5}{17} a^{14} - \frac{4}{17} a^{13} + \frac{5}{17} a^{12} - \frac{6}{17} a^{11} - \frac{2}{17} a^{10} + \frac{4}{17} a^{9} - \frac{5}{17} a^{8} - \frac{8}{17} a^{7} - \frac{3}{17} a^{6} - \frac{4}{17} a^{5} + \frac{7}{17} a^{4} + \frac{8}{17} a^{3} + \frac{7}{17} a^{2} + \frac{2}{17} a + \frac{4}{17}$, $\frac{1}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{27} + \frac{782280446429006099949731226285754944916077547068303562913547548245212122843}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{26} - \frac{18978122364051969015763927296646571041948277924753073551822030592440573981001}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{25} - \frac{16981071128675680288247566171408874692324597304457078027387934387302697798348}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{24} + \frac{110116560412917074703495660105269519704436456086085963373975574727933862839}{4708762439973107439235466247836249974210183521370729256377134540481103982563} a^{23} - \frac{9450533795632250578729240990261599554562154903121272918088690666972667174508}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{22} + \frac{4124263429211839558208086470858186661399105813095644902089174460500691406826}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{21} - \frac{1102331871990275933260857378026508109013156195391590406979021255902568475072}{4708762439973107439235466247836249974210183521370729256377134540481103982563} a^{20} + \frac{35997466673821916221154830315766968999961784470951970851223468828327239791966}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{19} - \frac{18510580233013392040220391096855868047203386070385270773389364488206831319115}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{18} - \frac{4454888853754101281070050317935560719779718780013192956994278485681416618911}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{17} - \frac{25659069215706848669102412901149134726313104028388633118941692617115492949640}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{16} - \frac{13989215477221820809003502157910088043765093044419407000849831761966390392691}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{15} + \frac{2170600460173448657798643176846181322666226576649564030600449200711092901563}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{14} - \frac{39080668365694672852216846746761834378631127753361743835943169439962409728996}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{13} - \frac{4327258033721295433307875430051288462072285130283610999986110573234516788506}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{12} - \frac{1652901953701183524969423236976412507891240972547645413063794963075990270119}{4708762439973107439235466247836249974210183521370729256377134540481103982563} a^{11} + \frac{11829185464852797020676093291816538715058419080328947181027731461014006617706}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{10} - \frac{20773402527717398380052863196618900614291388002435279556021957991634545997696}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{9} + \frac{29254728604901012419723765250661017478420899914883269449694530339296199960323}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{8} - \frac{23510328843977682379337468461805178573763581189833066163221047482388669100293}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{7} - \frac{6565048229968783124289635706446649464498968315055072098288197990550559433599}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{6} - \frac{14588626083290044688472498951748277998499460655537298956269364655512721537112}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{5} + \frac{2018409263002897730155168716452211185534414450002601654245477782704584471164}{4708762439973107439235466247836249974210183521370729256377134540481103982563} a^{4} + \frac{7544110733470880464417783415868543108853131618303306054540930226313681205724}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{3} - \frac{36114874427565559406650607918030851984551430123149415584948861270682790763127}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a^{2} + \frac{11213545184470313426582787591576570329898856228692293411536566710633927705389}{80048961479542826467002926213216249561573119863302397358411287188178767703571} a + \frac{107157662435619682996069993774130454561935166677046987551438470318124246256}{343557774590312559944218567438696350049670042331769945744254451451411020187}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $27$
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:  \( 694701292841794300 \) (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

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{377}) \), \(\Q(\sqrt{13}, \sqrt{29})\), 7.7.594823321.1, \(\Q(\zeta_{29})^+\), 14.14.22201352938819688612162197.1, 14.14.643839235225770969752703713.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
$13$13.14.7.1$x^{14} - 43940 x^{8} + 482680900 x^{2} - 250994068$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
13.14.7.1$x^{14} - 43940 x^{8} + 482680900 x^{2} - 250994068$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$