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Properties

Label 20.0.28787186638...8224.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 401^{9}$
Root discriminant $41.97$
Ramified primes $2, 401$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $C_5:D_4$ (as 20T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![191, 178, 523, 1314, 2221, -8320, 10322, -6508, 5673, -4260, 3429, -2872, 1356, -608, 449, -382, 290, -134, 42, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 42*x^18 - 134*x^17 + 290*x^16 - 382*x^15 + 449*x^14 - 608*x^13 + 1356*x^12 - 2872*x^11 + 3429*x^10 - 4260*x^9 + 5673*x^8 - 6508*x^7 + 10322*x^6 - 8320*x^5 + 2221*x^4 + 1314*x^3 + 523*x^2 + 178*x + 191)
 
gp: K = bnfinit(x^20 - 8*x^19 + 42*x^18 - 134*x^17 + 290*x^16 - 382*x^15 + 449*x^14 - 608*x^13 + 1356*x^12 - 2872*x^11 + 3429*x^10 - 4260*x^9 + 5673*x^8 - 6508*x^7 + 10322*x^6 - 8320*x^5 + 2221*x^4 + 1314*x^3 + 523*x^2 + 178*x + 191, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 42 x^{18} - 134 x^{17} + 290 x^{16} - 382 x^{15} + 449 x^{14} - 608 x^{13} + 1356 x^{12} - 2872 x^{11} + 3429 x^{10} - 4260 x^{9} + 5673 x^{8} - 6508 x^{7} + 10322 x^{6} - 8320 x^{5} + 2221 x^{4} + 1314 x^{3} + 523 x^{2} + 178 x + 191 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(287871866381166481585919674548224=2^{30}\cdot 401^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.97$
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:  $2, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{17} a^{17} + \frac{6}{17} a^{16} + \frac{1}{17} a^{15} + \frac{8}{17} a^{14} + \frac{3}{17} a^{13} - \frac{3}{17} a^{12} + \frac{1}{17} a^{11} + \frac{1}{17} a^{10} + \frac{5}{17} a^{9} + \frac{8}{17} a^{8} + \frac{3}{17} a^{7} + \frac{5}{17} a^{6} - \frac{1}{17} a^{5} - \frac{8}{17} a^{4} + \frac{3}{17} a^{3} + \frac{4}{17} a^{2} - \frac{5}{17} a - \frac{5}{17}$, $\frac{1}{51} a^{18} + \frac{16}{51} a^{16} - \frac{5}{17} a^{15} + \frac{2}{17} a^{14} - \frac{4}{51} a^{13} - \frac{5}{17} a^{12} + \frac{4}{17} a^{11} - \frac{6}{17} a^{10} - \frac{22}{51} a^{9} - \frac{11}{51} a^{8} + \frac{4}{51} a^{7} + \frac{1}{17} a^{6} + \frac{5}{17} a^{5} + \frac{1}{3} a^{4} + \frac{1}{17} a^{3} - \frac{4}{17} a^{2} - \frac{3}{17} a + \frac{13}{51}$, $\frac{1}{1816710300123752382412230563492349745719} a^{19} - \frac{501664479779328722212499937638767270}{78987404353206625322270894064884771553} a^{18} - \frac{884796302224551428224342648477583723}{58603558068508141368136469790075798249} a^{17} - \frac{196515816007901624306912477820206201432}{1816710300123752382412230563492349745719} a^{16} - \frac{5062265873480557757848862430563692115}{19534519356169380456045489930025266083} a^{15} - \frac{494853355009604906619132999636731996005}{1816710300123752382412230563492349745719} a^{14} - \frac{814998931035729443676408592623776360896}{1816710300123752382412230563492349745719} a^{13} + \frac{8549835127632975882801967981420086735}{26329134784402208440756964688294923851} a^{12} - \frac{184101225376425261456984087369838987374}{605570100041250794137410187830783248573} a^{11} + \frac{332524764460687433223868686982209673625}{1816710300123752382412230563492349745719} a^{10} + \frac{128088562510534301856726851517857331186}{605570100041250794137410187830783248573} a^{9} + \frac{404680431185875182268853766143487901247}{1816710300123752382412230563492349745719} a^{8} + \frac{267573310264995276321376083270870962473}{1816710300123752382412230563492349745719} a^{7} + \frac{136896138009581620609540197026183070262}{605570100041250794137410187830783248573} a^{6} + \frac{344683630803069061559463908770108974359}{1816710300123752382412230563492349745719} a^{5} + \frac{768187383737736684844384996030307885054}{1816710300123752382412230563492349745719} a^{4} + \frac{58219625043569299258497259525700148620}{605570100041250794137410187830783248573} a^{3} + \frac{5096886438911298481679171006714573769}{605570100041250794137410187830783248573} a^{2} + \frac{362336355562773612579960456404164631866}{1816710300123752382412230563492349745719} a + \frac{870330488552231364858868021998718321069}{1816710300123752382412230563492349745719}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  $9$
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:  \( 9087089.31038 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:D_4$ (as 20T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 13 conjugacy class representatives for $C_5:D_4$
Character table for $C_5:D_4$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.25664.1, 5.5.160801.1, 10.10.847280917741568.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 20 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
401Data not computed