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Properties

Label 28.0.17249157379...0000.1
Degree $28$
Signature $[0, 14]$
Discriminant $2^{28}\cdot 5^{14}\cdot 29^{26}$
Root discriminant $101.97$
Ramified primes $2, 5, 29$
Class number $3318784$ (GRH)
Class group $[2, 4, 4, 4, 25928]$ (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![83521, 0, 20075674, 0, 212892821, 0, 716818571, 0, 1154960464, 0, 1042796687, 0, 575107259, 0, 205287249, 0, 49145542, 0, 8023932, 0, 892864, 0, 66361, 0, 3138, 0, 85, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 85*x^26 + 3138*x^24 + 66361*x^22 + 892864*x^20 + 8023932*x^18 + 49145542*x^16 + 205287249*x^14 + 575107259*x^12 + 1042796687*x^10 + 1154960464*x^8 + 716818571*x^6 + 212892821*x^4 + 20075674*x^2 + 83521)
 
gp: K = bnfinit(x^28 + 85*x^26 + 3138*x^24 + 66361*x^22 + 892864*x^20 + 8023932*x^18 + 49145542*x^16 + 205287249*x^14 + 575107259*x^12 + 1042796687*x^10 + 1154960464*x^8 + 716818571*x^6 + 212892821*x^4 + 20075674*x^2 + 83521, 1)
 

Normalized defining polynomial

\( x^{28} + 85 x^{26} + 3138 x^{24} + 66361 x^{22} + 892864 x^{20} + 8023932 x^{18} + 49145542 x^{16} + 205287249 x^{14} + 575107259 x^{12} + 1042796687 x^{10} + 1154960464 x^{8} + 716818571 x^{6} + 212892821 x^{4} + 20075674 x^{2} + 83521 \)

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:  \(172491573797176117219488267129250761475686400000000000000=2^{28}\cdot 5^{14}\cdot 29^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $101.97$
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:  $2, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(580=2^{2}\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{580}(1,·)$, $\chi_{580}(579,·)$, $\chi_{580}(179,·)$, $\chi_{580}(129,·)$, $\chi_{580}(9,·)$, $\chi_{580}(209,·)$, $\chi_{580}(109,·)$, $\chi_{580}(141,·)$, $\chi_{580}(399,·)$, $\chi_{580}(401,·)$, $\chi_{580}(451,·)$, $\chi_{580}(149,·)$, $\chi_{580}(471,·)$, $\chi_{580}(281,·)$, $\chi_{580}(111,·)$, $\chi_{580}(161,·)$, $\chi_{580}(419,·)$, $\chi_{580}(81,·)$, $\chi_{580}(291,·)$, $\chi_{580}(299,·)$, $\chi_{580}(289,·)$, $\chi_{580}(431,·)$, $\chi_{580}(371,·)$, $\chi_{580}(181,·)$, $\chi_{580}(439,·)$, $\chi_{580}(571,·)$, $\chi_{580}(499,·)$, $\chi_{580}(469,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{17} a^{13} - \frac{2}{17} a^{11} - \frac{5}{17} a^{9} - \frac{6}{17} a^{7} + \frac{6}{17} a^{5} + \frac{8}{17} a^{3} - \frac{2}{17} a$, $\frac{1}{17} a^{14} - \frac{2}{17} a^{12} - \frac{5}{17} a^{10} - \frac{6}{17} a^{8} + \frac{6}{17} a^{6} + \frac{8}{17} a^{4} - \frac{2}{17} a^{2}$, $\frac{1}{17} a^{15} + \frac{8}{17} a^{11} + \frac{1}{17} a^{9} - \frac{6}{17} a^{7} + \frac{3}{17} a^{5} - \frac{3}{17} a^{3} - \frac{4}{17} a$, $\frac{1}{17} a^{16} + \frac{8}{17} a^{12} + \frac{1}{17} a^{10} - \frac{6}{17} a^{8} + \frac{3}{17} a^{6} - \frac{3}{17} a^{4} - \frac{4}{17} a^{2}$, $\frac{1}{17} a^{17} - \frac{1}{17} a$, $\frac{1}{17} a^{18} - \frac{1}{17} a^{2}$, $\frac{1}{17} a^{19} - \frac{1}{17} a^{3}$, $\frac{1}{1003} a^{20} - \frac{22}{1003} a^{18} - \frac{23}{1003} a^{16} - \frac{10}{1003} a^{14} + \frac{397}{1003} a^{12} - \frac{3}{17} a^{10} + \frac{215}{1003} a^{8} - \frac{231}{1003} a^{6} + \frac{481}{1003} a^{4} - \frac{342}{1003} a^{2} + \frac{9}{59}$, $\frac{1}{1003} a^{21} - \frac{22}{1003} a^{19} - \frac{23}{1003} a^{17} - \frac{10}{1003} a^{15} - \frac{16}{1003} a^{13} - \frac{6}{17} a^{11} + \frac{274}{1003} a^{9} + \frac{241}{1003} a^{7} + \frac{9}{1003} a^{5} + \frac{366}{1003} a^{3} - \frac{24}{1003} a$, $\frac{1}{1003} a^{22} + \frac{24}{1003} a^{18} + \frac{15}{1003} a^{16} + \frac{120}{1003} a^{12} - \frac{257}{1003} a^{10} + \frac{369}{1003} a^{8} - \frac{58}{1003} a^{6} + \frac{210}{1003} a^{4} + \frac{358}{1003} a^{2} + \frac{21}{59}$, $\frac{1}{1003} a^{23} + \frac{24}{1003} a^{19} + \frac{15}{1003} a^{17} + \frac{2}{1003} a^{13} - \frac{21}{1003} a^{11} - \frac{44}{1003} a^{9} - \frac{353}{1003} a^{7} - \frac{498}{1003} a^{5} + \frac{417}{1003} a^{3} - \frac{410}{1003} a$, $\frac{1}{1006009} a^{24} - \frac{234}{1006009} a^{22} + \frac{60}{1006009} a^{20} + \frac{8121}{1006009} a^{18} - \frac{29236}{1006009} a^{16} - \frac{3367}{1006009} a^{14} + \frac{12723}{59177} a^{12} - \frac{96551}{1006009} a^{10} + \frac{349263}{1006009} a^{8} + \frac{385426}{1006009} a^{6} - \frac{67043}{1006009} a^{4} + \frac{353676}{1006009} a^{2} - \frac{1155}{3481}$, $\frac{1}{1006009} a^{25} - \frac{234}{1006009} a^{23} + \frac{60}{1006009} a^{21} + \frac{8121}{1006009} a^{19} - \frac{29236}{1006009} a^{17} - \frac{3367}{1006009} a^{15} - \frac{1201}{59177} a^{13} + \frac{376865}{1006009} a^{11} - \frac{479215}{1006009} a^{9} - \frac{206344}{1006009} a^{7} - \frac{481282}{1006009} a^{5} + \frac{472030}{1006009} a^{3} + \frac{8213}{59177} a$, $\frac{1}{482916177686875456189098388807} a^{26} + \frac{202610477820393367937063}{482916177686875456189098388807} a^{24} + \frac{69444371149383228254916372}{482916177686875456189098388807} a^{22} + \frac{42965676895254228050998470}{482916177686875456189098388807} a^{20} - \frac{11288950448767253941761735158}{482916177686875456189098388807} a^{18} + \frac{3164898890749252660810375665}{482916177686875456189098388807} a^{16} + \frac{1126119382635924395934728275}{482916177686875456189098388807} a^{14} + \frac{127396720088744808630752237015}{482916177686875456189098388807} a^{12} - \frac{36041014967544925929448439096}{482916177686875456189098388807} a^{10} + \frac{46322683745232997397954749492}{482916177686875456189098388807} a^{8} - \frac{115072529421428379010569571627}{482916177686875456189098388807} a^{6} - \frac{18869537459172528287908772450}{482916177686875456189098388807} a^{4} - \frac{118776906239788786772398624190}{482916177686875456189098388807} a^{2} + \frac{267306246333538238397825711}{1670990234210641716917295463}$, $\frac{1}{8209575020676882755214672609719} a^{27} + \frac{68392577212668312817877}{482916177686875456189098388807} a^{25} + \frac{1289204838006390008687794215}{8209575020676882755214672609719} a^{23} + \frac{58828335826791310057756307}{139145339333506487376519874741} a^{21} + \frac{1679003547190402706972924195}{139145339333506487376519874741} a^{19} - \frac{13105200418328437751985985150}{482916177686875456189098388807} a^{17} - \frac{206250441123081537700753908263}{8209575020676882755214672609719} a^{15} + \frac{85165934149451155931585598590}{8209575020676882755214672609719} a^{13} + \frac{579508872484590147515519691457}{8209575020676882755214672609719} a^{11} - \frac{1315069206450731839917882115966}{8209575020676882755214672609719} a^{9} + \frac{358477270371187369858652267604}{8209575020676882755214672609719} a^{7} - \frac{2959547372189498538908157937834}{8209575020676882755214672609719} a^{5} - \frac{613616110481482949932135273471}{8209575020676882755214672609719} a^{3} + \frac{9410949446920812993882067015}{28406833981580909187594022871} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{25928}$, which has order $3318784$ (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:  \( -\frac{2851445306650935}{1974413521630346573813} a^{27} - \frac{14135141072529873}{116141971860608621989} a^{25} - \frac{8779109101798424672}{1974413521630346573813} a^{23} - \frac{183338186123211784183}{1974413521630346573813} a^{21} - \frac{2430101383215309245999}{1974413521630346573813} a^{19} - \frac{1262196929531605980398}{116141971860608621989} a^{17} - \frac{128759259837154320902823}{1974413521630346573813} a^{15} - \frac{525324567717087293316025}{1974413521630346573813} a^{13} - \frac{1432627620833301908927150}{1974413521630346573813} a^{11} - \frac{2518567154731074137005939}{1974413521630346573813} a^{9} - \frac{2687020991522342272081181}{1974413521630346573813} a^{7} - \frac{1581636339622852854829659}{1974413521630346573813} a^{5} - \frac{426122980706803485362735}{1974413521630346573813} a^{3} - \frac{109803253495073500389}{6831880697682860117} a \) (order $4$)
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:  \( 73869644668.60387 \) (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

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-145}) \), \(\Q(i, \sqrt{145})\), 7.7.594823321.1, 14.14.801611618199890796015625.1, 14.0.5796901408038404767744.1, 14.0.13133604752587010801920000000.1

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 ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{28}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
$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}$
$5$5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$