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Properties

Label 28.28.3539448906...7381.1
Degree $28$
Signature $[28, 0]$
Discriminant $23^{14}\cdot 29^{27}$
Root discriminant $123.32$
Ramified primes $23, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{28}$ (as 28T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-121377840329, 2393938599113, -2393938599113, -10862665827127, 10862665827127, 12115448511689, -12115448511689, -6485882143543, 6485882143543, 2039727740105, -2039727740105, -414614499127, 414614499127, 57374393033, -57374393033, -5557459255, 5557459255, 380963081, -380963081, -18404503, 18404503, 613001, -613001, -13399, 13399, 173, -173, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - 173*x^26 + 173*x^25 + 13399*x^24 - 13399*x^23 - 613001*x^22 + 613001*x^21 + 18404503*x^20 - 18404503*x^19 - 380963081*x^18 + 380963081*x^17 + 5557459255*x^16 - 5557459255*x^15 - 57374393033*x^14 + 57374393033*x^13 + 414614499127*x^12 - 414614499127*x^11 - 2039727740105*x^10 + 2039727740105*x^9 + 6485882143543*x^8 - 6485882143543*x^7 - 12115448511689*x^6 + 12115448511689*x^5 + 10862665827127*x^4 - 10862665827127*x^3 - 2393938599113*x^2 + 2393938599113*x - 121377840329)
 
gp: K = bnfinit(x^28 - x^27 - 173*x^26 + 173*x^25 + 13399*x^24 - 13399*x^23 - 613001*x^22 + 613001*x^21 + 18404503*x^20 - 18404503*x^19 - 380963081*x^18 + 380963081*x^17 + 5557459255*x^16 - 5557459255*x^15 - 57374393033*x^14 + 57374393033*x^13 + 414614499127*x^12 - 414614499127*x^11 - 2039727740105*x^10 + 2039727740105*x^9 + 6485882143543*x^8 - 6485882143543*x^7 - 12115448511689*x^6 + 12115448511689*x^5 + 10862665827127*x^4 - 10862665827127*x^3 - 2393938599113*x^2 + 2393938599113*x - 121377840329, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} - 173 x^{26} + 173 x^{25} + 13399 x^{24} - 13399 x^{23} - 613001 x^{22} + 613001 x^{21} + 18404503 x^{20} - 18404503 x^{19} - 380963081 x^{18} + 380963081 x^{17} + 5557459255 x^{16} - 5557459255 x^{15} - 57374393033 x^{14} + 57374393033 x^{13} + 414614499127 x^{12} - 414614499127 x^{11} - 2039727740105 x^{10} + 2039727740105 x^{9} + 6485882143543 x^{8} - 6485882143543 x^{7} - 12115448511689 x^{6} + 12115448511689 x^{5} + 10862665827127 x^{4} - 10862665827127 x^{3} - 2393938599113 x^{2} + 2393938599113 x - 121377840329 \)

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:  \(35394489068231220324814698212289719250778220848093751207381=23^{14}\cdot 29^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $123.32$
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:  $23, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(667=23\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{667}(576,·)$, $\chi_{667}(1,·)$, $\chi_{667}(323,·)$, $\chi_{667}(68,·)$, $\chi_{667}(645,·)$, $\chi_{667}(321,·)$, $\chi_{667}(137,·)$, $\chi_{667}(139,·)$, $\chi_{667}(206,·)$, $\chi_{667}(208,·)$, $\chi_{667}(275,·)$, $\chi_{667}(277,·)$, $\chi_{667}(24,·)$, $\chi_{667}(93,·)$, $\chi_{667}(415,·)$, $\chi_{667}(160,·)$, $\chi_{667}(482,·)$, $\chi_{667}(484,·)$, $\chi_{667}(229,·)$, $\chi_{667}(231,·)$, $\chi_{667}(298,·)$, $\chi_{667}(620,·)$, $\chi_{667}(622,·)$, $\chi_{667}(367,·)$, $\chi_{667}(114,·)$, $\chi_{667}(505,·)$, $\chi_{667}(254,·)$, $\chi_{667}(597,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{75943352179} a^{15} + \frac{23960563227}{75943352179} a^{14} - \frac{90}{75943352179} a^{13} + \frac{37783197765}{75943352179} a^{12} + \frac{3240}{75943352179} a^{11} - \frac{31751891381}{75943352179} a^{10} - \frac{59400}{75943352179} a^{9} - \frac{25834943051}{75943352179} a^{8} + \frac{583200}{75943352179} a^{7} + \frac{19560805963}{75943352179} a^{6} - \frac{2939328}{75943352179} a^{5} - \frac{2299871673}{75943352179} a^{4} + \frac{6531840}{75943352179} a^{3} - \frac{34521868580}{75943352179} a^{2} - \frac{4199040}{75943352179} a + \frac{13103928153}{75943352179}$, $\frac{1}{75943352179} a^{16} - \frac{96}{75943352179} a^{14} - \frac{8123324996}{75943352179} a^{13} + \frac{3744}{75943352179} a^{12} + \frac{26072532256}{75943352179} a^{11} - \frac{76032}{75943352179} a^{10} - \frac{22742445890}{75943352179} a^{9} + \frac{855360}{75943352179} a^{8} + \frac{23717812100}{75943352179} a^{7} - \frac{5225472}{75943352179} a^{6} - \frac{14137980342}{75943352179} a^{5} + \frac{15676416}{75943352179} a^{4} - \frac{33529411153}{75943352179} a^{3} - \frac{17915904}{75943352179} a^{2} + \frac{28739495274}{75943352179} a + \frac{3359232}{75943352179}$, $\frac{1}{75943352179} a^{17} + \frac{13790179426}{75943352179} a^{14} - \frac{4896}{75943352179} a^{13} + \frac{7978613104}{75943352179} a^{12} + \frac{235008}{75943352179} a^{11} - \frac{33189931306}{75943352179} a^{10} - \frac{4847040}{75943352179} a^{9} - \frac{26249451068}{75943352179} a^{8} + \frac{50761728}{75943352179} a^{7} - \frac{34884412369}{75943352179} a^{6} - \frac{266499072}{75943352179} a^{5} - \frac{26487035224}{75943352179} a^{4} + \frac{609140736}{75943352179} a^{3} - \frac{19795744709}{75943352179} a^{2} - \frac{399748608}{75943352179} a - \frac{33059884355}{75943352179}$, $\frac{1}{75943352179} a^{18} - \frac{5508}{75943352179} a^{14} + \frac{34001126580}{75943352179} a^{13} + \frac{286416}{75943352179} a^{12} + \frac{17263161885}{75943352179} a^{11} - \frac{6543504}{75943352179} a^{10} - \frac{14588149362}{75943352179} a^{9} + \frac{78522048}{75943352179} a^{8} + \frac{9413452710}{75943352179} a^{7} - \frac{499685760}{75943352179} a^{6} - \frac{18880484598}{75943352179} a^{5} + \frac{1541887488}{75943352179} a^{4} + \frac{5491560666}{75943352179} a^{3} - \frac{1798868736}{75943352179} a^{2} - \frac{33042433772}{75943352179} a + \frac{342641664}{75943352179}$, $\frac{1}{75943352179} a^{19} + \frac{19237293794}{75943352179} a^{14} - \frac{209304}{75943352179} a^{13} - \frac{33611871134}{75943352179} a^{12} + \frac{11302416}{75943352179} a^{11} - \frac{6465807673}{75943352179} a^{10} - \frac{248653152}{75943352179} a^{9} + \frac{28389111248}{75943352179} a^{8} + \frac{2712579840}{75943352179} a^{7} + \frac{34365369784}{75943352179} a^{6} - \frac{14647931136}{75943352179} a^{5} + \frac{20338199675}{75943352179} a^{4} + \frac{34178505984}{75943352179} a^{3} - \frac{17340716196}{75943352179} a^{2} - \frac{22785670656}{75943352179} a + \frac{30251696674}{75943352179}$, $\frac{1}{75943352179} a^{20} - \frac{246240}{75943352179} a^{14} + \frac{26990822388}{75943352179} a^{13} + \frac{14405040}{75943352179} a^{12} + \frac{14194438726}{75943352179} a^{11} - \frac{351039744}{75943352179} a^{10} + \frac{4020237435}{75943352179} a^{9} + \frac{4387996800}{75943352179} a^{8} + \frac{31985464833}{75943352179} a^{7} - \frac{28721433600}{75943352179} a^{6} - \frac{25384027028}{75943352179} a^{5} + \frac{14529163661}{75943352179} a^{4} + \frac{16759030096}{75943352179} a^{3} - \frac{31283333261}{75943352179} a^{2} - \frac{13367669422}{75943352179} a + \frac{20679432192}{75943352179}$, $\frac{1}{75943352179} a^{21} + \frac{37049052358}{75943352179} a^{14} - \frac{7756560}{75943352179} a^{13} + \frac{4679995215}{75943352179} a^{12} + \frac{446777856}{75943352179} a^{11} + \frac{14223464582}{75943352179} a^{10} - \frac{10238659200}{75943352179} a^{9} - \frac{17609435114}{75943352179} a^{8} - \frac{37000969958}{75943352179} a^{7} - \frac{3692298804}{75943352179} a^{6} - \frac{25760793448}{75943352179} a^{5} + \frac{5935469379}{75943352179} a^{4} - \frac{17693447420}{75943352179} a^{3} - \frac{35104004436}{75943352179} a^{2} - \frac{26028599081}{75943352179} a + \frac{30121013368}{75943352179}$, $\frac{1}{75943352179} a^{22} - \frac{9480240}{75943352179} a^{14} - \frac{2412788441}{75943352179} a^{13} + \frac{591566976}{75943352179} a^{12} - \frac{34209732518}{75943352179} a^{11} - \frac{15016700160}{75943352179} a^{10} + \frac{9641187024}{75943352179} a^{9} - \frac{34758197337}{75943352179} a^{8} + \frac{11817723781}{75943352179} a^{7} + \frac{965927843}{75943352179} a^{6} - \frac{26669033142}{75943352179} a^{5} + \frac{27286371774}{75943352179} a^{4} - \frac{13278455842}{75943352179} a^{3} - \frac{17554975693}{75943352179} a^{2} - \frac{5607797065}{75943352179} a - \frac{22223409367}{75943352179}$, $\frac{1}{75943352179} a^{23} + \frac{5112304509}{75943352179} a^{14} - \frac{261654624}{75943352179} a^{13} + \frac{17172629293}{75943352179} a^{12} + \frac{15699277440}{75943352179} a^{11} - \frac{15223004801}{75943352179} a^{10} + \frac{9662364095}{75943352179} a^{9} - \frac{36973795793}{75943352179} a^{8} - \frac{14022813224}{75943352179} a^{7} - \frac{26914370487}{75943352179} a^{6} + \frac{32961742747}{75943352179} a^{5} - \frac{12297106462}{75943352179} a^{4} + \frac{12023840022}{75943352179} a^{3} - \frac{7079821135}{75943352179} a^{2} - \frac{35813837171}{75943352179} a + \frac{20508731983}{75943352179}$, $\frac{1}{75943352179} a^{24} - \frac{330511104}{75943352179} a^{14} + \frac{21619922029}{75943352179} a^{13} + \frac{21483221760}{75943352179} a^{12} - \frac{23438838939}{75943352179} a^{11} - \frac{29321094107}{75943352179} a^{10} + \frac{12392027165}{75943352179} a^{9} - \frac{4370319763}{75943352179} a^{8} + \frac{13102528253}{75943352179} a^{7} - \frac{2553191018}{75943352179} a^{6} - \frac{31718230882}{75943352179} a^{5} + \frac{2196870724}{75943352179} a^{4} - \frac{14550668321}{75943352179} a^{3} - \frac{21073383224}{75943352179} a^{2} + \frac{36160469771}{75943352179} a - \frac{28407736372}{75943352179}$, $\frac{1}{75943352179} a^{25} + \frac{32156257966}{75943352179} a^{14} - \frac{8262777600}{75943352179} a^{13} + \frac{2829168501}{75943352179} a^{12} - \frac{21672047653}{75943352179} a^{11} - \frac{19075344386}{75943352179} a^{10} + \frac{32598316998}{75943352179} a^{9} - \frac{3580311620}{75943352179} a^{8} + \frac{7294831480}{75943352179} a^{7} - \frac{28280409871}{75943352179} a^{6} - \frac{10984353620}{75943352179} a^{5} - \frac{28301952365}{75943352179} a^{4} - \frac{17096224297}{75943352179} a^{3} - \frac{19155956342}{75943352179} a^{2} + \frac{7007194693}{75943352179} a - \frac{23649864060}{75943352179}$, $\frac{1}{75943352179} a^{26} - \frac{10741610880}{75943352179} a^{14} + \frac{11045002639}{75943352179} a^{13} + \frac{34663243509}{75943352179} a^{12} - \frac{11071964638}{75943352179} a^{11} - \frac{3825839052}{75943352179} a^{10} + \frac{26892214751}{75943352179} a^{9} - \frac{24932164819}{75943352179} a^{8} - \frac{30595746632}{75943352179} a^{7} + \frac{1515017883}{75943352179} a^{6} + \frac{31971090305}{75943352179} a^{5} + \frac{8849641870}{75943352179} a^{4} + \frac{15666953678}{75943352179} a^{3} + \frac{6060071532}{75943352179} a^{2} + \frac{8209231055}{75943352179} a - \frac{35414931065}{75943352179}$, $\frac{1}{75943352179} a^{27} + \frac{11730835786}{75943352179} a^{14} - \frac{20761509543}{75943352179} a^{13} + \frac{7261403193}{75943352179} a^{12} + \frac{16938114166}{75943352179} a^{11} + \frac{29437893760}{75943352179} a^{10} - \frac{573428861}{75943352179} a^{9} - \frac{3212901162}{75943352179} a^{8} + \frac{17802340352}{75943352179} a^{7} + \frac{29785244894}{75943352179} a^{6} + \frac{36119963764}{75943352179} a^{5} + \frac{24410335322}{75943352179} a^{4} + \frac{21402708391}{75943352179} a^{3} - \frac{4905841411}{75943352179} a^{2} + \frac{14391721952}{75943352179} a - \frac{9187060160}{75943352179}$

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:  \( 260881845825591150000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{28}$ (as 28T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 28
The 28 conjugacy class representatives for $C_{28}$
Character table for $C_{28}$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), 4.4.12901781.1, 7.7.594823321.1, \(\Q(\zeta_{29})^+\)

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 $28$ $28$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ $28$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ $28$ R R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

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
$23$23.14.7.2$x^{14} - 148035889 x^{2} + 27238603576$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
23.14.7.2$x^{14} - 148035889 x^{2} + 27238603576$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
29Data not computed