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Properties

Label 20.0.36528267636...5625.6
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 11^{16}\cdot 13^{10}$
Root discriminant $95.09$
Ramified primes $3, 5, 11, 13$
Class number $180620$ (GRH)
Class group $[2, 90310]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![523826161, -136713220, 425478541, -101338586, 143264265, -31392896, 27779377, -5804132, 3759952, -802500, 404716, -65800, 30574, -4148, 2228, -972, 249, 38, -5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 5*x^18 + 38*x^17 + 249*x^16 - 972*x^15 + 2228*x^14 - 4148*x^13 + 30574*x^12 - 65800*x^11 + 404716*x^10 - 802500*x^9 + 3759952*x^8 - 5804132*x^7 + 27779377*x^6 - 31392896*x^5 + 143264265*x^4 - 101338586*x^3 + 425478541*x^2 - 136713220*x + 523826161)
 
gp: K = bnfinit(x^20 - 4*x^19 - 5*x^18 + 38*x^17 + 249*x^16 - 972*x^15 + 2228*x^14 - 4148*x^13 + 30574*x^12 - 65800*x^11 + 404716*x^10 - 802500*x^9 + 3759952*x^8 - 5804132*x^7 + 27779377*x^6 - 31392896*x^5 + 143264265*x^4 - 101338586*x^3 + 425478541*x^2 - 136713220*x + 523826161, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 5 x^{18} + 38 x^{17} + 249 x^{16} - 972 x^{15} + 2228 x^{14} - 4148 x^{13} + 30574 x^{12} - 65800 x^{11} + 404716 x^{10} - 802500 x^{9} + 3759952 x^{8} - 5804132 x^{7} + 27779377 x^{6} - 31392896 x^{5} + 143264265 x^{4} - 101338586 x^{3} + 425478541 x^{2} - 136713220 x + 523826161 \)

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:  \(3652826763616533724927636470896103515625=3^{10}\cdot 5^{10}\cdot 11^{16}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.09$
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:  $3, 5, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2145=3\cdot 5\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2145}(1,·)$, $\chi_{2145}(961,·)$, $\chi_{2145}(1156,·)$, $\chi_{2145}(389,·)$, $\chi_{2145}(1351,·)$, $\chi_{2145}(584,·)$, $\chi_{2145}(586,·)$, $\chi_{2145}(779,·)$, $\chi_{2145}(1741,·)$, $\chi_{2145}(14,·)$, $\chi_{2145}(1169,·)$, $\chi_{2145}(1171,·)$, $\chi_{2145}(599,·)$, $\chi_{2145}(196,·)$, $\chi_{2145}(1754,·)$, $\chi_{2145}(1951,·)$, $\chi_{2145}(1379,·)$, $\chi_{2145}(1574,·)$, $\chi_{2145}(1769,·)$, $\chi_{2145}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{46} a^{14} + \frac{3}{23} a^{13} + \frac{1}{23} a^{12} + \frac{1}{23} a^{11} + \frac{1}{46} a^{10} - \frac{1}{46} a^{9} - \frac{11}{23} a^{8} - \frac{1}{46} a^{7} - \frac{11}{23} a^{6} + \frac{11}{23} a^{5} + \frac{1}{46} a^{4} + \frac{7}{46} a^{2} + \frac{1}{46} a - \frac{10}{23}$, $\frac{1}{322} a^{15} - \frac{3}{322} a^{14} + \frac{17}{322} a^{13} - \frac{8}{161} a^{12} + \frac{3}{161} a^{11} - \frac{4}{23} a^{10} - \frac{18}{161} a^{9} + \frac{15}{46} a^{8} + \frac{79}{322} a^{7} + \frac{151}{322} a^{6} + \frac{33}{322} a^{5} + \frac{53}{161} a^{4} + \frac{61}{161} a^{3} - \frac{8}{161} a^{2} + \frac{66}{161} a - \frac{48}{161}$, $\frac{1}{322} a^{16} + \frac{1}{322} a^{14} - \frac{1}{46} a^{13} - \frac{4}{23} a^{12} - \frac{26}{161} a^{11} - \frac{25}{161} a^{10} - \frac{157}{322} a^{9} + \frac{65}{322} a^{8} + \frac{73}{322} a^{7} - \frac{2}{161} a^{6} + \frac{51}{322} a^{5} + \frac{111}{322} a^{4} - \frac{19}{46} a^{3} - \frac{9}{23} a^{2} + \frac{66}{161} a + \frac{13}{322}$, $\frac{1}{322} a^{17} + \frac{3}{322} a^{14} - \frac{31}{322} a^{13} - \frac{11}{161} a^{12} - \frac{3}{23} a^{11} + \frac{67}{322} a^{10} - \frac{67}{322} a^{9} - \frac{25}{322} a^{8} - \frac{45}{161} a^{7} + \frac{34}{161} a^{6} - \frac{45}{161} a^{5} + \frac{45}{161} a^{4} - \frac{87}{322} a^{3} + \frac{18}{161} a^{2} + \frac{7}{46} a + \frac{117}{322}$, $\frac{1}{42131317473964695914} a^{18} - \frac{3452283937640}{56476296881990209} a^{17} - \frac{24331564370351705}{21065658736982347957} a^{16} + \frac{41011986675814897}{42131317473964695914} a^{15} - \frac{163737887989423763}{21065658736982347957} a^{14} - \frac{9351718676947602401}{42131317473964695914} a^{13} - \frac{498835021954384140}{3009379819568906851} a^{12} - \frac{1845895256200833599}{42131317473964695914} a^{11} - \frac{7884474907874986175}{42131317473964695914} a^{10} + \frac{521258516020858391}{21065658736982347957} a^{9} + \frac{529661153634137758}{3009379819568906851} a^{8} - \frac{18218302865763841131}{42131317473964695914} a^{7} + \frac{5711472021902830397}{42131317473964695914} a^{6} - \frac{2457120594784070666}{21065658736982347957} a^{5} - \frac{9062690857033817111}{21065658736982347957} a^{4} - \frac{1119122188877435573}{21065658736982347957} a^{3} + \frac{16881027888127586387}{42131317473964695914} a^{2} + \frac{2748093179130761880}{21065658736982347957} a + \frac{368938792695123657}{21065658736982347957}$, $\frac{1}{13016534188375461751768288088295003284153287394} a^{19} + \frac{14709853155866189040973504}{6508267094187730875884144044147501642076643697} a^{18} + \frac{4879787399310793755122668301035026041270259}{6508267094187730875884144044147501642076643697} a^{17} + \frac{14952008984964030617233973237917525389443659}{13016534188375461751768288088295003284153287394} a^{16} - \frac{2443580318669647003455036061684571992800715}{6508267094187730875884144044147501642076643697} a^{15} - \frac{34554509288291244859217963792469770731885703}{6508267094187730875884144044147501642076643697} a^{14} + \frac{1449161096295699026869471368435898998128685497}{13016534188375461751768288088295003284153287394} a^{13} + \frac{1203962127765520853228946103978964732813753707}{13016534188375461751768288088295003284153287394} a^{12} + \frac{2332045225729313930697689983553324393172273983}{13016534188375461751768288088295003284153287394} a^{11} - \frac{417757885188495128339655616380474988568239544}{6508267094187730875884144044147501642076643697} a^{10} + \frac{5236048604346585580553915241822733580633780725}{13016534188375461751768288088295003284153287394} a^{9} - \frac{2536717848936244102738717449634548154233604091}{13016534188375461751768288088295003284153287394} a^{8} + \frac{841103784055442673330064062205173413685830}{282968134529901342429745393223804419220723639} a^{7} - \frac{3288939274922731225421875525998330962033838639}{13016534188375461751768288088295003284153287394} a^{6} + \frac{1791228175881599418090850857393999556257122726}{6508267094187730875884144044147501642076643697} a^{5} + \frac{1048079272415265694606860146071117302295933271}{13016534188375461751768288088295003284153287394} a^{4} + \frac{1999001360174232765548206360012737799656455513}{13016534188375461751768288088295003284153287394} a^{3} - \frac{35747178915985600444015965094433830851259179}{282968134529901342429745393223804419220723639} a^{2} - \frac{95171986539585272735535625345815208094198683}{565936269059802684859490786447608838441447278} a + \frac{453192741569241597155023760594377076961381965}{6508267094187730875884144044147501642076643697}$

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

Class group and class number

$C_{2}\times C_{90310}$, which has order $180620$ (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:  \( 2015201.7242 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-195}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{13}, \sqrt{-15})\), \(\Q(\zeta_{11})^+\), 10.0.60438619802379121875.1, 10.10.79589952003133.1, 10.0.162778775259375.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5Data not computed
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
13Data not computed