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Properties

Label 28.0.75862596433...7856.3
Degree $28$
Signature $[0, 14]$
Discriminant $2^{56}\cdot 29^{26}$
Root discriminant $91.20$
Ramified primes $2, 29$
Class number $16512$ (GRH)
Class group $[2, 4, 4, 516]$ (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![11042329, 0, 20759960, 0, 17488517, 0, -8370860, 0, 684982, 0, -1054280, 0, 55130, 0, 92448, 0, -401, 0, -4584, 0, 1294, 0, 188, 0, -63, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 4*x^26 - 63*x^24 + 188*x^22 + 1294*x^20 - 4584*x^18 - 401*x^16 + 92448*x^14 + 55130*x^12 - 1054280*x^10 + 684982*x^8 - 8370860*x^6 + 17488517*x^4 + 20759960*x^2 + 11042329)
 
gp: K = bnfinit(x^28 - 4*x^26 - 63*x^24 + 188*x^22 + 1294*x^20 - 4584*x^18 - 401*x^16 + 92448*x^14 + 55130*x^12 - 1054280*x^10 + 684982*x^8 - 8370860*x^6 + 17488517*x^4 + 20759960*x^2 + 11042329, 1)
 

Normalized defining polynomial

\( x^{28} - 4 x^{26} - 63 x^{24} + 188 x^{22} + 1294 x^{20} - 4584 x^{18} - 401 x^{16} + 92448 x^{14} + 55130 x^{12} - 1054280 x^{10} + 684982 x^{8} - 8370860 x^{6} + 17488517 x^{4} + 20759960 x^{2} + 11042329 \)

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:  \(7586259643335085676646037106440640773336778620436217856=2^{56}\cdot 29^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.20$
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:  $2, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(232=2^{3}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{232}(1,·)$, $\chi_{232}(107,·)$, $\chi_{232}(5,·)$, $\chi_{232}(65,·)$, $\chi_{232}(139,·)$, $\chi_{232}(109,·)$, $\chi_{232}(13,·)$, $\chi_{232}(207,·)$, $\chi_{232}(81,·)$, $\chi_{232}(83,·)$, $\chi_{232}(149,·)$, $\chi_{232}(151,·)$, $\chi_{232}(25,·)$, $\chi_{232}(219,·)$, $\chi_{232}(93,·)$, $\chi_{232}(231,·)$, $\chi_{232}(161,·)$, $\chi_{232}(227,·)$, $\chi_{232}(167,·)$, $\chi_{232}(169,·)$, $\chi_{232}(71,·)$, $\chi_{232}(173,·)$, $\chi_{232}(49,·)$, $\chi_{232}(123,·)$, $\chi_{232}(183,·)$, $\chi_{232}(59,·)$, $\chi_{232}(125,·)$, $\chi_{232}(63,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{14947} a^{24} + \frac{5184}{14947} a^{22} - \frac{2083}{14947} a^{20} + \frac{6867}{14947} a^{18} + \frac{5596}{14947} a^{16} - \frac{1504}{14947} a^{14} - \frac{6448}{14947} a^{12} + \frac{5364}{14947} a^{10} - \frac{6802}{14947} a^{8} + \frac{2631}{14947} a^{6} - \frac{1594}{14947} a^{4} - \frac{3575}{14947} a^{2} + \frac{4626}{14947}$, $\frac{1}{49668881} a^{25} - \frac{20023796}{49668881} a^{23} + \frac{23748700}{49668881} a^{21} + \frac{18630829}{49668881} a^{19} - \frac{233556}{49668881} a^{17} - \frac{4246452}{49668881} a^{15} + \frac{13834474}{49668881} a^{13} + \frac{21723355}{49668881} a^{11} - \frac{7614825}{49668881} a^{9} + \frac{1168497}{49668881} a^{7} - \frac{11137109}{49668881} a^{5} - \frac{3232127}{49668881} a^{3} + \frac{18628588}{49668881} a$, $\frac{1}{2370770751647005520703394125369730532841602640107} a^{26} + \frac{12023614138798410852010985976045854889142179}{2370770751647005520703394125369730532841602640107} a^{24} - \frac{706137895797102281179880112182408503722400501756}{2370770751647005520703394125369730532841602640107} a^{22} + \frac{391872708948767182263900349856799463082171648977}{2370770751647005520703394125369730532841602640107} a^{20} + \frac{1059267035064473249788206915545195113315017875214}{2370770751647005520703394125369730532841602640107} a^{18} - \frac{783935310046315299886763557514858764710661893946}{2370770751647005520703394125369730532841602640107} a^{16} + \frac{604642556747457307272275760823082705335608465988}{2370770751647005520703394125369730532841602640107} a^{14} - \frac{1156997576343948167978344271823732643754332555057}{2370770751647005520703394125369730532841602640107} a^{12} - \frac{1101850884633599886236254961932241200726324906704}{2370770751647005520703394125369730532841602640107} a^{10} + \frac{257575355828409872381623231875169189633437035685}{2370770751647005520703394125369730532841602640107} a^{8} + \frac{666847656031300479827870122559915452677889665900}{2370770751647005520703394125369730532841602640107} a^{6} - \frac{1156530004886609875544272045110978326318727941529}{2370770751647005520703394125369730532841602640107} a^{4} + \frac{82996793716329940400642353357975494258208574783}{2370770751647005520703394125369730532841602640107} a^{2} + \frac{157796150735127978184738046032777875457858967}{713442898479387758261629288404974581053747409}$, $\frac{1}{2370770751647005520703394125369730532841602640107} a^{27} - \frac{4726693261916643936734682150010128958065}{2370770751647005520703394125369730532841602640107} a^{25} + \frac{699059227501597599825298350174536845251326253661}{2370770751647005520703394125369730532841602640107} a^{23} - \frac{773095011761669936342438201071676151282776220983}{2370770751647005520703394125369730532841602640107} a^{21} - \frac{186243505950687065092921822314744458881884777004}{2370770751647005520703394125369730532841602640107} a^{19} - \frac{345414890320638971946591836838995847384843946389}{2370770751647005520703394125369730532841602640107} a^{17} - \frac{474541996502422332380204712873841619337832282078}{2370770751647005520703394125369730532841602640107} a^{15} + \frac{761957286316476627788791824353553687582099880884}{2370770751647005520703394125369730532841602640107} a^{13} + \frac{757784592342137301107095655285412624418542146553}{2370770751647005520703394125369730532841602640107} a^{11} - \frac{608773481911022110720645696497993754733875411188}{2370770751647005520703394125369730532841602640107} a^{9} + \frac{836391988673337106015811128838404963672424693274}{2370770751647005520703394125369730532841602640107} a^{7} + \frac{41250838687642863132480206063203890420914567075}{2370770751647005520703394125369730532841602640107} a^{5} + \frac{1027789935868891737641354875254913582149853672059}{2370770751647005520703394125369730532841602640107} a^{3} - \frac{694198420317682849454825368898001758544874658073}{2370770751647005520703394125369730532841602640107} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{516}$, which has order $16512$ (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:  \( -1 \) (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:  \( 306576699955.91235 \) (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

\(\Q(\sqrt{58}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-29}) \), \(\Q(\sqrt{-2}, \sqrt{-29})\), 7.7.594823321.1, 14.14.21518098026638558497865728.1, 14.0.742003380228915810271232.1, 14.0.168110140833113738264576.1

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 ${\href{/LocalNumberField/3.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
29Data not computed