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Properties

Label 28.0.20501251744...8064.1
Degree $28$
Signature $[0, 14]$
Discriminant $2^{28}\cdot 3^{14}\cdot 43^{24}$
Root discriminant $87.04$
Ramified primes $2, 3, 43$
Class number $8729$ (GRH)
Class group $[8729]$ (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![5764801, 0, -24588641, 0, 71950767, 0, -99291934, 0, 93616839, 0, -61574116, 0, 30588386, 0, -11374748, 0, 3252533, 0, -689424, 0, 109211, 0, -11816, 0, 899, 0, -37, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 37*x^26 + 899*x^24 - 11816*x^22 + 109211*x^20 - 689424*x^18 + 3252533*x^16 - 11374748*x^14 + 30588386*x^12 - 61574116*x^10 + 93616839*x^8 - 99291934*x^6 + 71950767*x^4 - 24588641*x^2 + 5764801)
 
gp: K = bnfinit(x^28 - 37*x^26 + 899*x^24 - 11816*x^22 + 109211*x^20 - 689424*x^18 + 3252533*x^16 - 11374748*x^14 + 30588386*x^12 - 61574116*x^10 + 93616839*x^8 - 99291934*x^6 + 71950767*x^4 - 24588641*x^2 + 5764801, 1)
 

Normalized defining polynomial

\( x^{28} - 37 x^{26} + 899 x^{24} - 11816 x^{22} + 109211 x^{20} - 689424 x^{18} + 3252533 x^{16} - 11374748 x^{14} + 30588386 x^{12} - 61574116 x^{10} + 93616839 x^{8} - 99291934 x^{6} + 71950767 x^{4} - 24588641 x^{2} + 5764801 \)

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:  \(2050125174461338816008034482151201300557313248344408064=2^{28}\cdot 3^{14}\cdot 43^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.04$
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:  $2, 3, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(516=2^{2}\cdot 3\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{516}(1,·)$, $\chi_{516}(259,·)$, $\chi_{516}(133,·)$, $\chi_{516}(385,·)$, $\chi_{516}(11,·)$, $\chi_{516}(269,·)$, $\chi_{516}(365,·)$, $\chi_{516}(145,·)$, $\chi_{516}(403,·)$, $\chi_{516}(355,·)$, $\chi_{516}(47,·)$, $\chi_{516}(107,·)$, $\chi_{516}(97,·)$, $\chi_{516}(35,·)$, $\chi_{516}(391,·)$, $\chi_{516}(293,·)$, $\chi_{516}(379,·)$, $\chi_{516}(41,·)$, $\chi_{516}(193,·)$, $\chi_{516}(299,·)$, $\chi_{516}(173,·)$, $\chi_{516}(451,·)$, $\chi_{516}(431,·)$, $\chi_{516}(305,·)$, $\chi_{516}(121,·)$, $\chi_{516}(59,·)$, $\chi_{516}(317,·)$, $\chi_{516}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} + \frac{1}{7} a$, $\frac{1}{7} a^{8} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} + \frac{1}{7} a^{4}$, $\frac{1}{7} a^{11} + \frac{1}{7} a^{5}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{6}$, $\frac{1}{7} a^{13} - \frac{1}{7} a$, $\frac{1}{49} a^{14} + \frac{2}{49} a^{8} + \frac{1}{49} a^{2}$, $\frac{1}{49} a^{15} + \frac{2}{49} a^{9} + \frac{1}{49} a^{3}$, $\frac{1}{49} a^{16} + \frac{2}{49} a^{10} + \frac{1}{49} a^{4}$, $\frac{1}{343} a^{17} - \frac{3}{343} a^{15} - \frac{3}{49} a^{13} + \frac{2}{343} a^{11} - \frac{13}{343} a^{9} - \frac{3}{49} a^{7} + \frac{148}{343} a^{5} - \frac{10}{343} a^{3} + \frac{1}{7} a$, $\frac{1}{343} a^{18} - \frac{3}{343} a^{16} + \frac{2}{343} a^{12} - \frac{13}{343} a^{10} + \frac{3}{49} a^{8} + \frac{148}{343} a^{6} - \frac{10}{343} a^{4} + \frac{10}{49} a^{2}$, $\frac{1}{343} a^{19} - \frac{2}{343} a^{15} - \frac{12}{343} a^{13} - \frac{1}{49} a^{11} - \frac{4}{343} a^{9} - \frac{13}{343} a^{7} + \frac{13}{49} a^{5} + \frac{47}{343} a^{3}$, $\frac{1}{88837} a^{20} + \frac{104}{88837} a^{18} - \frac{328}{88837} a^{16} - \frac{726}{88837} a^{14} + \frac{5787}{88837} a^{12} - \frac{4569}{88837} a^{10} + \frac{253}{88837} a^{8} - \frac{6714}{88837} a^{6} - \frac{31289}{88837} a^{4} - \frac{898}{1813} a^{2} + \frac{7}{37}$, $\frac{1}{88837} a^{21} + \frac{104}{88837} a^{19} - \frac{69}{88837} a^{17} + \frac{310}{88837} a^{15} + \frac{348}{88837} a^{13} - \frac{4051}{88837} a^{11} + \frac{512}{88837} a^{9} + \frac{538}{88837} a^{7} + \frac{7043}{88837} a^{5} + \frac{6294}{12691} a^{3} + \frac{123}{259} a$, $\frac{1}{88837} a^{22} - \frac{1}{12691} a^{18} - \frac{25}{88837} a^{16} - \frac{6}{1813} a^{14} - \frac{51}{12691} a^{12} + \frac{682}{88837} a^{10} - \frac{8}{1813} a^{8} - \frac{4656}{12691} a^{6} + \frac{14771}{88837} a^{4} + \frac{236}{1813} a^{2} + \frac{12}{37}$, $\frac{1}{88837} a^{23} - \frac{1}{12691} a^{19} - \frac{25}{88837} a^{17} - \frac{6}{1813} a^{15} - \frac{51}{12691} a^{13} + \frac{682}{88837} a^{11} - \frac{8}{1813} a^{9} + \frac{783}{12691} a^{7} + \frac{14771}{88837} a^{5} + \frac{236}{1813} a^{3} - \frac{64}{259} a$, $\frac{1}{88837} a^{24} - \frac{2}{2401} a^{18} - \frac{1}{343} a^{16} + \frac{1564}{88837} a^{12} + \frac{12}{343} a^{10} + \frac{1}{49} a^{8} - \frac{206}{2401} a^{6} + \frac{13}{343} a^{4} - \frac{13}{49} a^{2} + \frac{12}{37}$, $\frac{1}{621859} a^{25} - \frac{2}{621859} a^{23} + \frac{3}{621859} a^{21} + \frac{36}{88837} a^{19} + \frac{620}{621859} a^{17} + \frac{2036}{621859} a^{15} + \frac{19639}{621859} a^{13} + \frac{4248}{88837} a^{11} - \frac{40156}{621859} a^{9} + \frac{37420}{621859} a^{7} - \frac{168993}{621859} a^{5} - \frac{4609}{12691} a^{3} + \frac{9}{259} a$, $\frac{1}{902956550091201613224077515459} a^{26} + \frac{4289749219319939240447601}{902956550091201613224077515459} a^{24} - \frac{896762801448407592080603}{902956550091201613224077515459} a^{22} - \frac{631089683759252350669821}{128993792870171659032011073637} a^{20} - \frac{723583415317712580057962503}{902956550091201613224077515459} a^{18} - \frac{7846646896650653354614865662}{902956550091201613224077515459} a^{16} - \frac{7304690337493270708931193105}{902956550091201613224077515459} a^{14} + \frac{4342944371134663276850951848}{128993792870171659032011073637} a^{12} + \frac{55980646458915917797701972780}{902956550091201613224077515459} a^{10} - \frac{33813813187041459680029580640}{902956550091201613224077515459} a^{8} - \frac{47202781118199081781927157272}{902956550091201613224077515459} a^{6} - \frac{328149174445090278310631506}{2632526385105544061877777013} a^{4} + \frac{13740389722312626411933855}{53725028267460082895464837} a^{2} + \frac{1907846384410477391376126}{7675004038208583270780691}$, $\frac{1}{6320695850638411292568542608213} a^{27} + \frac{4289749219319939240447601}{6320695850638411292568542608213} a^{25} - \frac{31389346412709535721939024}{6320695850638411292568542608213} a^{23} + \frac{2272965898265616995030981}{902956550091201613224077515459} a^{21} - \frac{3661035636535867923234323726}{6320695850638411292568542608213} a^{19} + \frac{2043114387935039202169215549}{6320695850638411292568542608213} a^{17} + \frac{42184772863583540245829024178}{6320695850638411292568542608213} a^{15} + \frac{2771850301259208960826817966}{902956550091201613224077515459} a^{13} + \frac{139743773639050236770423055267}{6320695850638411292568542608213} a^{11} + \frac{241269947764682264222133188008}{6320695850638411292568542608213} a^{9} - \frac{98094903165393904630660861921}{6320695850638411292568542608213} a^{7} - \frac{2966098438851762412538033486}{18427684695738808433144439091} a^{5} + \frac{97424603636988453271720231}{376075197872220580268253859} a^{3} + \frac{20369342584425718231902653}{53725028267460082895464837} a$

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

Class group and class number

$C_{8729}$, which has order $8729$ (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:  \( -\frac{702552625440833448228524}{6320695850638411292568542608213} a^{27} + \frac{22590158191712233281183282}{6320695850638411292568542608213} a^{25} - \frac{515036051543751530787087088}{6320695850638411292568542608213} a^{23} + \frac{797567647984299180487112568}{902956550091201613224077515459} a^{21} - \frac{44689632286000558212613656425}{6320695850638411292568542608213} a^{19} + \frac{216803629100634460009253538070}{6320695850638411292568542608213} a^{17} - \frac{858104776069595042494620099634}{6320695850638411292568542608213} a^{15} + \frac{333101226173592726650367465235}{902956550091201613224077515459} a^{13} - \frac{5834671658562542783606958636278}{6320695850638411292568542608213} a^{11} + \frac{9815814069286565660186201865648}{6320695850638411292568542608213} a^{9} - \frac{15177569450807493323062597822543}{6320695850638411292568542608213} a^{7} + \frac{527747664298346291515043926}{376075197872220580268253859} a^{5} - \frac{26359579654724058992997294}{53725028267460082895464837} a^{3} - \frac{80027787959442893570331775}{53725028267460082895464837} a \) (order $12$)
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:  \( 3767000207683.561 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), 7.7.6321363049.1, 14.14.1431825818478399563185963008.1, 14.0.654698590982350051753984.1, 14.0.87391712553613254588987.1

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 ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{28}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$

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
2Data not computed
3Data not computed
$43$43.14.12.1$x^{14} + 3569 x^{7} + 4043763$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$
43.14.12.1$x^{14} + 3569 x^{7} + 4043763$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$