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Properties

Label 20.20.1716317724...4813.1
Degree $20$
Signature $[20, 0]$
Discriminant $13^{12}\cdot 277^{11}$
Root discriminant $102.74$
Ramified primes $13, 277$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T138

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -102, -2850, 3203, 75997, -134595, -306986, 519154, 554743, -663530, -540165, 289218, 224656, -43042, -37137, 2334, 2754, -22, -90, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 90*x^18 - 22*x^17 + 2754*x^16 + 2334*x^15 - 37137*x^14 - 43042*x^13 + 224656*x^12 + 289218*x^11 - 540165*x^10 - 663530*x^9 + 554743*x^8 + 519154*x^7 - 306986*x^6 - 134595*x^5 + 75997*x^4 + 3203*x^3 - 2850*x^2 - 102*x + 1)
 
gp: K = bnfinit(x^20 - x^19 - 90*x^18 - 22*x^17 + 2754*x^16 + 2334*x^15 - 37137*x^14 - 43042*x^13 + 224656*x^12 + 289218*x^11 - 540165*x^10 - 663530*x^9 + 554743*x^8 + 519154*x^7 - 306986*x^6 - 134595*x^5 + 75997*x^4 + 3203*x^3 - 2850*x^2 - 102*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 90 x^{18} - 22 x^{17} + 2754 x^{16} + 2334 x^{15} - 37137 x^{14} - 43042 x^{13} + 224656 x^{12} + 289218 x^{11} - 540165 x^{10} - 663530 x^{9} + 554743 x^{8} + 519154 x^{7} - 306986 x^{6} - 134595 x^{5} + 75997 x^{4} + 3203 x^{3} - 2850 x^{2} - 102 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17163177241103084972955413201910386914813=13^{12}\cdot 277^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $102.74$
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:  $13, 277$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{277} a^{16} + \frac{53}{277} a^{15} - \frac{122}{277} a^{14} + \frac{122}{277} a^{13} - \frac{63}{277} a^{12} - \frac{44}{277} a^{11} - \frac{8}{277} a^{10} + \frac{11}{277} a^{9} - \frac{103}{277} a^{8} + \frac{45}{277} a^{7} - \frac{21}{277} a^{6} + \frac{83}{277} a^{5} - \frac{110}{277} a^{4} - \frac{108}{277} a^{3} + \frac{105}{277} a^{2} - \frac{98}{277} a - \frac{41}{277}$, $\frac{1}{1939} a^{17} - \frac{3}{1939} a^{16} - \frac{43}{1939} a^{15} + \frac{860}{1939} a^{14} + \frac{123}{277} a^{13} - \frac{117}{1939} a^{12} + \frac{517}{1939} a^{11} + \frac{736}{1939} a^{10} + \frac{666}{1939} a^{9} + \frac{827}{1939} a^{8} - \frac{879}{1939} a^{7} - \frac{957}{1939} a^{6} - \frac{880}{1939} a^{5} - \frac{596}{1939} a^{4} + \frac{59}{1939} a^{3} - \frac{23}{277} a^{2} - \frac{132}{277} a + \frac{80}{1939}$, $\frac{1}{32963} a^{18} + \frac{8}{32963} a^{17} - \frac{2}{1939} a^{16} - \frac{14838}{32963} a^{15} + \frac{1319}{32963} a^{14} - \frac{14607}{32963} a^{13} - \frac{1042}{4709} a^{12} + \frac{12331}{32963} a^{11} - \frac{7086}{32963} a^{10} + \frac{8615}{32963} a^{9} - \frac{829}{4709} a^{8} + \frac{137}{4709} a^{7} - \frac{6472}{32963} a^{6} - \frac{1247}{4709} a^{5} - \frac{11117}{32963} a^{4} - \frac{9865}{32963} a^{3} - \frac{32}{4709} a^{2} + \frac{12946}{32963} a + \frac{14670}{32963}$, $\frac{1}{190130872209088862821608375289122577109} a^{19} + \frac{2741688410438708185561232043105729}{190130872209088862821608375289122577109} a^{18} + \frac{60320095081282950998387185836153}{2290733400109504371344679220350874423} a^{17} - \frac{14335898532129162445372935339323910}{27161553172726980403086910755588939587} a^{16} - \frac{11572238336808745299637384886750763248}{190130872209088862821608375289122577109} a^{15} + \frac{59715309539828359722955898747944170814}{190130872209088862821608375289122577109} a^{14} - \frac{2419246043797432996898064278621304555}{11184168953475815460094610311124857477} a^{13} + \frac{14989819217308755678570276195816764865}{190130872209088862821608375289122577109} a^{12} + \frac{434302374322629132614936709048250044}{4637338346563142995648984763149331149} a^{11} + \frac{6538743029977458519429622873444322434}{190130872209088862821608375289122577109} a^{10} + \frac{73244419873786757526579967567857633409}{190130872209088862821608375289122577109} a^{9} + \frac{3170326930481642492177005012368679681}{11184168953475815460094610311124857477} a^{8} + \frac{61411779211797356291918168640262051081}{190130872209088862821608375289122577109} a^{7} + \frac{32238396186941460309378844027583136754}{190130872209088862821608375289122577109} a^{6} + \frac{35511125298872254012102361890976548552}{190130872209088862821608375289122577109} a^{5} - \frac{65066473762967991665247301865698023783}{190130872209088862821608375289122577109} a^{4} - \frac{65992599598333176722826228980880071395}{190130872209088862821608375289122577109} a^{3} - \frac{73607170595293505865193779210031748763}{190130872209088862821608375289122577109} a^{2} - \frac{25536062896983803818241699387122439221}{190130872209088862821608375289122577109} a - \frac{77137269771063830518100737726617148458}{190130872209088862821608375289122577109}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $19$
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:  \( 40323641683000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T138:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 28 conjugacy class representatives for t20n138
Character table for t20n138 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.12967201.1, 10.10.2185927923067213.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: 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 $20$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ $20$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
277Data not computed