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Properties

Label 28.0.13517157994...4944.1
Degree $28$
Signature $[0, 14]$
Discriminant $2^{28}\cdot 3^{14}\cdot 29^{26}$
Root discriminant $78.98$
Ramified primes $2, 3, 29$
Class number $384$ (GRH)
Class group $[2, 4, 4, 12]$ (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![841, 0, -10933, 0, 96715, 0, -459186, 0, 1563419, 0, -2610464, 0, 3056194, 0, -1923396, 0, 854369, 0, -232000, 0, 45675, 0, -5916, 0, 551, 0, -29, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 29*x^26 + 551*x^24 - 5916*x^22 + 45675*x^20 - 232000*x^18 + 854369*x^16 - 1923396*x^14 + 3056194*x^12 - 2610464*x^10 + 1563419*x^8 - 459186*x^6 + 96715*x^4 - 10933*x^2 + 841)
 
gp: K = bnfinit(x^28 - 29*x^26 + 551*x^24 - 5916*x^22 + 45675*x^20 - 232000*x^18 + 854369*x^16 - 1923396*x^14 + 3056194*x^12 - 2610464*x^10 + 1563419*x^8 - 459186*x^6 + 96715*x^4 - 10933*x^2 + 841, 1)
 

Normalized defining polynomial

\( x^{28} - 29 x^{26} + 551 x^{24} - 5916 x^{22} + 45675 x^{20} - 232000 x^{18} + 854369 x^{16} - 1923396 x^{14} + 3056194 x^{12} - 2610464 x^{10} + 1563419 x^{8} - 459186 x^{6} + 96715 x^{4} - 10933 x^{2} + 841 \)

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:  \(135171579942192030712001098144632895138136441638354944=2^{28}\cdot 3^{14}\cdot 29^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.98$
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:  $2, 3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(348=2^{2}\cdot 3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{348}(1,·)$, $\chi_{348}(323,·)$, $\chi_{348}(197,·)$, $\chi_{348}(257,·)$, $\chi_{348}(115,·)$, $\chi_{348}(67,·)$, $\chi_{348}(277,·)$, $\chi_{348}(151,·)$, $\chi_{348}(25,·)$, $\chi_{348}(71,·)$, $\chi_{348}(283,·)$, $\chi_{348}(161,·)$, $\chi_{348}(35,·)$, $\chi_{348}(65,·)$, $\chi_{348}(167,·)$, $\chi_{348}(169,·)$, $\chi_{348}(299,·)$, $\chi_{348}(281,·)$, $\chi_{348}(49,·)$, $\chi_{348}(91,·)$, $\chi_{348}(179,·)$, $\chi_{348}(53,·)$, $\chi_{348}(233,·)$, $\chi_{348}(313,·)$, $\chi_{348}(347,·)$, $\chi_{348}(187,·)$, $\chi_{348}(295,·)$, $\chi_{348}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{29} a^{14}$, $\frac{1}{29} a^{15}$, $\frac{1}{29} a^{16}$, $\frac{1}{29} a^{17}$, $\frac{1}{29} a^{18}$, $\frac{1}{29} a^{19}$, $\frac{1}{493} a^{20} + \frac{5}{493} a^{18} + \frac{5}{493} a^{16} + \frac{7}{493} a^{14} + \frac{7}{17} a^{12} + \frac{7}{17} a^{10} + \frac{7}{17} a^{8} + \frac{5}{17} a^{6} - \frac{5}{17} a^{4} - \frac{5}{17} a^{2} + \frac{3}{17}$, $\frac{1}{493} a^{21} + \frac{5}{493} a^{19} + \frac{5}{493} a^{17} + \frac{7}{493} a^{15} + \frac{7}{17} a^{13} + \frac{7}{17} a^{11} + \frac{7}{17} a^{9} + \frac{5}{17} a^{7} - \frac{5}{17} a^{5} - \frac{5}{17} a^{3} + \frac{3}{17} a$, $\frac{1}{493} a^{22} - \frac{3}{493} a^{18} - \frac{1}{493} a^{16} - \frac{2}{493} a^{14} + \frac{6}{17} a^{12} + \frac{6}{17} a^{10} + \frac{4}{17} a^{8} + \frac{4}{17} a^{6} + \frac{3}{17} a^{4} - \frac{6}{17} a^{2} + \frac{2}{17}$, $\frac{1}{493} a^{23} - \frac{3}{493} a^{19} - \frac{1}{493} a^{17} - \frac{2}{493} a^{15} + \frac{6}{17} a^{13} + \frac{6}{17} a^{11} + \frac{4}{17} a^{9} + \frac{4}{17} a^{7} + \frac{3}{17} a^{5} - \frac{6}{17} a^{3} + \frac{2}{17} a$, $\frac{1}{1192567} a^{24} + \frac{75}{1192567} a^{22} - \frac{46}{1192567} a^{20} - \frac{6697}{1192567} a^{18} + \frac{3958}{1192567} a^{16} + \frac{10518}{1192567} a^{14} + \frac{4847}{41123} a^{12} + \frac{1076}{2419} a^{10} + \frac{11495}{41123} a^{8} - \frac{1119}{41123} a^{6} - \frac{14662}{41123} a^{4} + \frac{13163}{41123} a^{2} - \frac{7153}{41123}$, $\frac{1}{1192567} a^{25} + \frac{75}{1192567} a^{23} - \frac{46}{1192567} a^{21} - \frac{6697}{1192567} a^{19} + \frac{3958}{1192567} a^{17} + \frac{10518}{1192567} a^{15} + \frac{4847}{41123} a^{13} + \frac{1076}{2419} a^{11} + \frac{11495}{41123} a^{9} - \frac{1119}{41123} a^{7} - \frac{14662}{41123} a^{5} + \frac{13163}{41123} a^{3} - \frac{7153}{41123} a$, $\frac{1}{4148797675963432116791257813} a^{26} + \frac{730783171372417513830}{4148797675963432116791257813} a^{24} + \frac{148172050719390875156370}{4148797675963432116791257813} a^{22} + \frac{418083742714354067818507}{4148797675963432116791257813} a^{20} + \frac{14601585867910114087102059}{4148797675963432116791257813} a^{18} - \frac{15061120271947858903718923}{4148797675963432116791257813} a^{16} + \frac{21320472958448858331990261}{4148797675963432116791257813} a^{14} - \frac{66993927239283321874225162}{143061988826325245406595097} a^{12} + \frac{1122902977505565796330547}{8415411107430896788623241} a^{10} + \frac{32905080252152390412532207}{143061988826325245406595097} a^{8} + \frac{8663447507497570322530813}{143061988826325245406595097} a^{6} + \frac{41591129011249661476173452}{143061988826325245406595097} a^{4} + \frac{70166325602009033148484648}{143061988826325245406595097} a^{2} + \frac{1361671935242583618846075}{8415411107430896788623241}$, $\frac{1}{4148797675963432116791257813} a^{27} + \frac{730783171372417513830}{4148797675963432116791257813} a^{25} + \frac{148172050719390875156370}{4148797675963432116791257813} a^{23} + \frac{418083742714354067818507}{4148797675963432116791257813} a^{21} + \frac{14601585867910114087102059}{4148797675963432116791257813} a^{19} - \frac{15061120271947858903718923}{4148797675963432116791257813} a^{17} + \frac{21320472958448858331990261}{4148797675963432116791257813} a^{15} - \frac{66993927239283321874225162}{143061988826325245406595097} a^{13} + \frac{1122902977505565796330547}{8415411107430896788623241} a^{11} + \frac{32905080252152390412532207}{143061988826325245406595097} a^{9} + \frac{8663447507497570322530813}{143061988826325245406595097} a^{7} + \frac{41591129011249661476173452}{143061988826325245406595097} a^{5} + \frac{70166325602009033148484648}{143061988826325245406595097} a^{3} + \frac{1361671935242583618846075}{8415411107430896788623241} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{12}$, which has order $384$ (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:  \( \frac{17794974015976247830}{100887524644686236821031} a^{26} - \frac{513845098838430797670}{100887524644686236821031} a^{24} + \frac{9741588466385237087800}{100887524644686236821031} a^{22} - \frac{104075676595349242494679}{100887524644686236821031} a^{20} + \frac{800053248285444887451775}{100887524644686236821031} a^{18} - \frac{139003485903995192324975}{3478880160161594373139} a^{16} + \frac{14718196606973246016914140}{100887524644686236821031} a^{14} - \frac{1119818527282558012165915}{3478880160161594373139} a^{12} + \frac{1745638968351759372561100}{3478880160161594373139} a^{10} - \frac{1404909169814334023239958}{3478880160161594373139} a^{8} + \frac{815179770710311150035865}{3478880160161594373139} a^{6} - \frac{202696972803554977446945}{3478880160161594373139} a^{4} + \frac{49449697077842951634753}{3478880160161594373139} a^{2} - \frac{2088700679450172287950}{3478880160161594373139} \) (order $6$)
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:  \( 4155697700585.085 \) (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

\(\Q(\sqrt{87}) \), \(\Q(\sqrt{-29}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-29})\), 7.7.594823321.1, 14.14.367656878002019745584627712.1, 14.0.168110140833113738264576.1, 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 R R ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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

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
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
29Data not computed