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Properties

Label 20.0.17186024261...1408.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 7^{15}\cdot 11^{13}$
Root discriminant $40.90$
Ramified primes $2, 7, 11$
Class number $44$ (GRH)
Class group $[2, 22]$ (GRH)
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2248377097, -1765189036, 3221261412, -2034072024, 1945306784, -999145718, 653189606, -274849484, 135712461, -46979228, 18451103, -5256998, 1692149, -393036, 104804, -19266, 4226, -572, 100, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 100*x^18 - 572*x^17 + 4226*x^16 - 19266*x^15 + 104804*x^14 - 393036*x^13 + 1692149*x^12 - 5256998*x^11 + 18451103*x^10 - 46979228*x^9 + 135712461*x^8 - 274849484*x^7 + 653189606*x^6 - 999145718*x^5 + 1945306784*x^4 - 2034072024*x^3 + 3221261412*x^2 - 1765189036*x + 2248377097)
 
gp: K = bnfinit(x^20 - 8*x^19 + 100*x^18 - 572*x^17 + 4226*x^16 - 19266*x^15 + 104804*x^14 - 393036*x^13 + 1692149*x^12 - 5256998*x^11 + 18451103*x^10 - 46979228*x^9 + 135712461*x^8 - 274849484*x^7 + 653189606*x^6 - 999145718*x^5 + 1945306784*x^4 - 2034072024*x^3 + 3221261412*x^2 - 1765189036*x + 2248377097, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 100 x^{18} - 572 x^{17} + 4226 x^{16} - 19266 x^{15} + 104804 x^{14} - 393036 x^{13} + 1692149 x^{12} - 5256998 x^{11} + 18451103 x^{10} - 46979228 x^{9} + 135712461 x^{8} - 274849484 x^{7} + 653189606 x^{6} - 999145718 x^{5} + 1945306784 x^{4} - 2034072024 x^{3} + 3221261412 x^{2} - 1765189036 x + 2248377097 \)

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:  \(171860242615100874270814694801408=2^{20}\cdot 7^{15}\cdot 11^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.90$
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:  $2, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a^{19} - \frac{151118652480709911958694631618145658704735566923855033980936198503663745963156}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a^{18} + \frac{12266713856523096693958025787874024641700205892162272414414193269985801031907}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a^{17} + \frac{276495810353985860484041022732385826520060565971109760964641748131289581128291}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a^{16} - \frac{173069464535037102234312618025093673182833154910762361392066574833114501323128}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a^{15} - \frac{58364041512955551882571927839662637862209253921749043181708338731303991811461}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a^{14} - \frac{293431502942411741348036412349888938780198470059475318159979345962580914138915}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a^{13} + \frac{184761748959709863001850971314821653159904765488036944815106406918996679440946}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a^{12} - \frac{103861619998059601108700451033813623031990407602803670965438042948364524762133}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a^{11} - \frac{196723320632430706356454536795344350578929483665151983210774920729004949366386}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a^{10} + \frac{11101110622193988155979003577457477336207450160414001521210263519021114985617}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a^{9} - \frac{120498235286493345796153481248095111345747595610105352271410716358345366974350}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a^{8} - \frac{40302745837483620801829087074687273807615586556388704598892560401908756515367}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a^{7} + \frac{129182555962220400644454428076291210373450886407499080563947390730617307105131}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a^{6} + \frac{302359295561048871172136564016304257264190917617968300857581953751198073912008}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a^{5} + \frac{226168987588328938388669636215631607632374704069494504484714013510105620575063}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a^{4} - \frac{148429608923242789553624975792155323741381032362395065900172223476339763175050}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a^{3} + \frac{180173112179777159700030111243064479804585017485785144875561988046491023828609}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a^{2} + \frac{96382317825855502909269156010438628112003810138007568689139153194567924055037}{631771062367569349044499480881348151291048598269343645005836032176927373214821} a + \frac{268322623792118969890661687750930371252286543675459410260407129367075689634704}{631771062367569349044499480881348151291048598269343645005836032176927373214821}$

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

Class group and class number

$C_{2}\times C_{22}$, which has order $44$ (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:  \( 1071550.255200949 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.60368.1, 10.0.246071287.1

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 R $20$ $20$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ $20$

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$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
7Data not computed
$11$11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$