Properties

Label 28.0.482...544.1
Degree $28$
Signature $[0, 14]$
Discriminant $4.828\times 10^{50}$
Root discriminant $64.59$
Ramified primes $2, 7$
Class number $71$ (GRH)
Class group $[71]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 14*x^24 + 406*x^22 + 637*x^20 + 1736*x^18 + 25613*x^16 + 45672*x^14 + 146510*x^12 + 216314*x^10 + 175987*x^8 + 135485*x^6 + 138712*x^4 + 36281*x^2 + 6241)
 
gp: K = bnfinit(x^28 + 14*x^24 + 406*x^22 + 637*x^20 + 1736*x^18 + 25613*x^16 + 45672*x^14 + 146510*x^12 + 216314*x^10 + 175987*x^8 + 135485*x^6 + 138712*x^4 + 36281*x^2 + 6241, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6241, 0, 36281, 0, 138712, 0, 135485, 0, 175987, 0, 216314, 0, 146510, 0, 45672, 0, 25613, 0, 1736, 0, 637, 0, 406, 0, 14, 0, 0, 0, 1]);
 

\( x^{28} + 14 x^{24} + 406 x^{22} + 637 x^{20} + 1736 x^{18} + 25613 x^{16} + 45672 x^{14} + 146510 x^{12} + 216314 x^{10} + 175987 x^{8} + 135485 x^{6} + 138712 x^{4} + 36281 x^{2} + 6241 \)

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(482771783823117526797953812068649122826362283884544\)\(\medspace = 2^{28}\cdot 7^{50}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $64.59$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $28$
This field is Galois and abelian over $\Q$.
Conductor:  \(196=2^{2}\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{196}(1,·)$, $\chi_{196}(195,·)$, $\chi_{196}(69,·)$, $\chi_{196}(71,·)$, $\chi_{196}(139,·)$, $\chi_{196}(13,·)$, $\chi_{196}(15,·)$, $\chi_{196}(141,·)$, $\chi_{196}(83,·)$, $\chi_{196}(85,·)$, $\chi_{196}(153,·)$, $\chi_{196}(155,·)$, $\chi_{196}(29,·)$, $\chi_{196}(97,·)$, $\chi_{196}(27,·)$, $\chi_{196}(167,·)$, $\chi_{196}(41,·)$, $\chi_{196}(43,·)$, $\chi_{196}(125,·)$, $\chi_{196}(111,·)$, $\chi_{196}(99,·)$, $\chi_{196}(113,·)$, $\chi_{196}(181,·)$, $\chi_{196}(55,·)$, $\chi_{196}(57,·)$, $\chi_{196}(183,·)$, $\chi_{196}(169,·)$, $\chi_{196}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{91991} a^{24} + \frac{3060}{91991} a^{22} - \frac{34560}{91991} a^{20} + \frac{34418}{91991} a^{18} - \frac{19428}{91991} a^{16} + \frac{14963}{91991} a^{14} + \frac{32934}{91991} a^{12} + \frac{17885}{91991} a^{10} + \frac{35419}{91991} a^{8} + \frac{30248}{91991} a^{6} + \frac{24092}{91991} a^{4} + \frac{36749}{91991} a^{2} + \frac{37627}{91991}$, $\frac{1}{7267289} a^{25} - \frac{1008841}{7267289} a^{23} + \frac{2909152}{7267289} a^{21} - \frac{241555}{7267289} a^{19} - \frac{387392}{7267289} a^{17} - \frac{353001}{7267289} a^{15} - \frac{2174850}{7267289} a^{13} + \frac{3329561}{7267289} a^{11} - \frac{700509}{7267289} a^{9} - \frac{3189437}{7267289} a^{7} + \frac{760020}{7267289} a^{5} - \frac{1987053}{7267289} a^{3} - \frac{2906085}{7267289} a$, $\frac{1}{26634927568365291857549826889315883279} a^{26} + \frac{47001196400232933009111897385151}{26634927568365291857549826889315883279} a^{24} - \frac{7435606132843811697279887164238530644}{26634927568365291857549826889315883279} a^{22} + \frac{8532444037477742027388154485251021796}{26634927568365291857549826889315883279} a^{20} + \frac{11204774469908874811931812480723428561}{26634927568365291857549826889315883279} a^{18} + \frac{395429380637123973978516959203083011}{26634927568365291857549826889315883279} a^{16} + \frac{4171777298446071791722166468291812495}{26634927568365291857549826889315883279} a^{14} - \frac{1545733717954098413263144660185478097}{26634927568365291857549826889315883279} a^{12} + \frac{6433348929258755624731916218878392008}{26634927568365291857549826889315883279} a^{10} - \frac{4435286959587915413367783605040767415}{26634927568365291857549826889315883279} a^{8} + \frac{9127434553683963223575889495482528907}{26634927568365291857549826889315883279} a^{6} - \frac{10041516235186151286518638638539144288}{26634927568365291857549826889315883279} a^{4} + \frac{426064685386426805373844662418165096}{26634927568365291857549826889315883279} a^{2} + \frac{118971732640236254139819471009824805}{337150981878041669082909201130580801}$, $\frac{1}{26634927568365291857549826889315883279} a^{27} - \frac{644364191722168012486080837692}{26634927568365291857549826889315883279} a^{25} - \frac{12638666276425819342649614994559142239}{26634927568365291857549826889315883279} a^{23} + \frac{3098903992096835267953487388890279055}{26634927568365291857549826889315883279} a^{21} - \frac{3921129709666702618345914778973613853}{26634927568365291857549826889315883279} a^{19} - \frac{7781989178889497388612425950409204812}{26634927568365291857549826889315883279} a^{17} - \frac{5644219735398477450102552510382268941}{26634927568365291857549826889315883279} a^{15} - \frac{4463496538001714386640089279498912663}{26634927568365291857549826889315883279} a^{13} + \frac{7604113969339888657458091585146329759}{26634927568365291857549826889315883279} a^{11} + \frac{2305929476756668590620967632548876393}{26634927568365291857549826889315883279} a^{9} + \frac{1280382981215713615299319028716938624}{26634927568365291857549826889315883279} a^{7} + \frac{7016759940446716550146119731167485410}{26634927568365291857549826889315883279} a^{5} - \frac{11439391477148581274556135473210516341}{26634927568365291857549826889315883279} a^{3} - \frac{11948749578741247315902662582388440424}{26634927568365291857549826889315883279} a$

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

Class group and class number

$C_{71}$, which has order $71$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{1694604370159091923560}{48166563384628033225653779} a^{27} + \frac{1303749658382307238053}{48166563384628033225653779} a^{25} + \frac{23091480306941228567945}{48166563384628033225653779} a^{23} + \frac{706348121813264898059271}{48166563384628033225653779} a^{21} + \frac{1599630340084744788309013}{48166563384628033225653779} a^{19} + \frac{3516446537302038602278212}{48166563384628033225653779} a^{17} + \frac{45295872275776396519772716}{48166563384628033225653779} a^{15} + \frac{1636185412934435323346617}{718903931113851242173937} a^{13} + \frac{291545865208818568790686328}{48166563384628033225653779} a^{11} + \frac{530706564397956963731430086}{48166563384628033225653779} a^{9} + \frac{484268213751188869019094535}{48166563384628033225653779} a^{7} + \frac{319918831668409030699320007}{48166563384628033225653779} a^{5} + \frac{300713830725918321539129708}{48166563384628033225653779} a^{3} + \frac{131750254101061389231655791}{48166563384628033225653779} a \) (order $4$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 983258026102.4221 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 983258026102.4221 \cdot 71}{4\sqrt{482771783823117526797953812068649122826362283884544}}\approx 0.118717044151947$ (assuming GRH)

Galois group

$C_2\times C_{14}$ (as 28T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{7}) \), \(\Q(i, \sqrt{7})\), 7.7.13841287201.1, 14.0.1341068619663964900807.1, 14.0.3138866894939200133545984.1, 14.14.21972068264574400934821888.1

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 ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.14.14.38$x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.38$x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
7Data not computed