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Properties

Label 20.0.12902360962...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{34}\cdot 17^{10}$
Root discriminant $254.41$
Ramified primes $2, 5, 17$
Class number $417640400$ (GRH)
Class group $[10, 41764040]$ (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![208310963449, 19495942660, 89131121320, 5197770460, 18523918365, 960854012, 2477895450, 132916040, 229742855, 12226060, 15681528, 768720, 813850, 21440, 31670, 116, 1005, -20, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 20*x^18 - 20*x^17 + 1005*x^16 + 116*x^15 + 31670*x^14 + 21440*x^13 + 813850*x^12 + 768720*x^11 + 15681528*x^10 + 12226060*x^9 + 229742855*x^8 + 132916040*x^7 + 2477895450*x^6 + 960854012*x^5 + 18523918365*x^4 + 5197770460*x^3 + 89131121320*x^2 + 19495942660*x + 208310963449)
 
gp: K = bnfinit(x^20 + 20*x^18 - 20*x^17 + 1005*x^16 + 116*x^15 + 31670*x^14 + 21440*x^13 + 813850*x^12 + 768720*x^11 + 15681528*x^10 + 12226060*x^9 + 229742855*x^8 + 132916040*x^7 + 2477895450*x^6 + 960854012*x^5 + 18523918365*x^4 + 5197770460*x^3 + 89131121320*x^2 + 19495942660*x + 208310963449, 1)
 

Normalized defining polynomial

\( x^{20} + 20 x^{18} - 20 x^{17} + 1005 x^{16} + 116 x^{15} + 31670 x^{14} + 21440 x^{13} + 813850 x^{12} + 768720 x^{11} + 15681528 x^{10} + 12226060 x^{9} + 229742855 x^{8} + 132916040 x^{7} + 2477895450 x^{6} + 960854012 x^{5} + 18523918365 x^{4} + 5197770460 x^{3} + 89131121320 x^{2} + 19495942660 x + 208310963449 \)

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:  \(1290236096287360000000000000000000000000000000000=2^{40}\cdot 5^{34}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $254.41$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3400=2^{3}\cdot 5^{2}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{3400}(1,·)$, $\chi_{3400}(3331,·)$, $\chi_{3400}(69,·)$, $\chi_{3400}(3399,·)$, $\chi_{3400}(1291,·)$, $\chi_{3400}(1359,·)$, $\chi_{3400}(1361,·)$, $\chi_{3400}(1429,·)$, $\chi_{3400}(2651,·)$, $\chi_{3400}(2719,·)$, $\chi_{3400}(2721,·)$, $\chi_{3400}(611,·)$, $\chi_{3400}(2789,·)$, $\chi_{3400}(679,·)$, $\chi_{3400}(681,·)$, $\chi_{3400}(749,·)$, $\chi_{3400}(1971,·)$, $\chi_{3400}(2039,·)$, $\chi_{3400}(2041,·)$, $\chi_{3400}(2109,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{25} a^{10} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{8}{25} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{7}{25}$, $\frac{1}{25} a^{11} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{8}{25} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{7}{25} a$, $\frac{1}{175} a^{12} + \frac{1}{175} a^{11} - \frac{2}{175} a^{10} - \frac{3}{7} a^{9} + \frac{17}{35} a^{8} - \frac{82}{175} a^{7} - \frac{87}{175} a^{6} + \frac{54}{175} a^{5} - \frac{17}{35} a^{3} + \frac{62}{175} a^{2} + \frac{2}{175} a + \frac{41}{175}$, $\frac{1}{175} a^{13} - \frac{3}{175} a^{11} - \frac{3}{175} a^{10} - \frac{3}{35} a^{9} + \frac{8}{175} a^{8} - \frac{1}{35} a^{7} - \frac{34}{175} a^{6} - \frac{19}{175} a^{5} - \frac{17}{35} a^{4} - \frac{4}{25} a^{3} - \frac{12}{35} a^{2} + \frac{39}{175} a - \frac{76}{175}$, $\frac{1}{175} a^{14} - \frac{6}{25} a^{9} - \frac{13}{35} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{25} a^{4} - \frac{4}{35} a^{2} + \frac{2}{5} a - \frac{16}{35}$, $\frac{1}{1925} a^{15} - \frac{3}{1925} a^{13} - \frac{4}{1925} a^{12} + \frac{19}{1925} a^{11} - \frac{4}{1925} a^{10} + \frac{7}{55} a^{9} - \frac{37}{275} a^{8} - \frac{21}{275} a^{7} - \frac{628}{1925} a^{6} + \frac{46}{175} a^{5} - \frac{2}{7} a^{4} + \frac{789}{1925} a^{3} + \frac{142}{1925} a^{2} + \frac{208}{1925} a - \frac{314}{1925}$, $\frac{1}{1925} a^{16} - \frac{3}{1925} a^{14} - \frac{4}{1925} a^{13} - \frac{3}{1925} a^{12} - \frac{26}{1925} a^{11} - \frac{19}{1925} a^{10} - \frac{534}{1925} a^{9} + \frac{678}{1925} a^{8} + \frac{58}{275} a^{7} - \frac{1}{7} a^{6} - \frac{32}{175} a^{5} - \frac{366}{1925} a^{4} - \frac{683}{1925} a^{3} - \frac{771}{1925} a^{2} + \frac{797}{1925} a + \frac{72}{175}$, $\frac{1}{1925} a^{17} - \frac{4}{1925} a^{14} - \frac{1}{1925} a^{13} - \frac{1}{385} a^{12} + \frac{38}{1925} a^{11} - \frac{29}{1925} a^{10} + \frac{698}{1925} a^{9} - \frac{943}{1925} a^{8} - \frac{12}{1925} a^{7} - \frac{13}{275} a^{6} - \frac{47}{1925} a^{5} - \frac{958}{1925} a^{4} + \frac{23}{1925} a^{3} - \frac{86}{1925} a^{2} - \frac{57}{275} a + \frac{37}{1925}$, $\frac{1}{853710479935580575} a^{18} - \frac{130917120310464}{853710479935580575} a^{17} + \frac{33350810894152}{170742095987116115} a^{16} - \frac{31110022863776}{170742095987116115} a^{15} + \frac{242555338604659}{121958639990797225} a^{14} + \frac{1435313272380308}{853710479935580575} a^{13} + \frac{31719088995362}{17422662855828175} a^{12} - \frac{535780752583012}{853710479935580575} a^{11} + \frac{16491353279082959}{853710479935580575} a^{10} - \frac{346128805008949}{1583878441438925} a^{9} + \frac{151257943733161217}{853710479935580575} a^{8} + \frac{36902219578626112}{77610043630507325} a^{7} + \frac{14452989145723004}{77610043630507325} a^{6} + \frac{29780896796824767}{77610043630507325} a^{5} - \frac{18661532751828269}{77610043630507325} a^{4} - \frac{32104514642065746}{170742095987116115} a^{3} - \frac{20392433534011668}{170742095987116115} a^{2} + \frac{69705248346221386}{853710479935580575} a - \frac{396105215879905437}{853710479935580575}$, $\frac{1}{8678530342864570245312849318820645112295832022960575} a^{19} + \frac{331602983561527223522577551687749}{1735706068572914049062569863764129022459166404592115} a^{18} - \frac{144981651443716417583243664307557125573590974582}{8678530342864570245312849318820645112295832022960575} a^{17} + \frac{2215755694262937125375109425643252299215952627678}{8678530342864570245312849318820645112295832022960575} a^{16} - \frac{1679324574749803453390933286948768416943414147431}{8678530342864570245312849318820645112295832022960575} a^{15} - \frac{20396857452772526866869465936144126002245993423933}{8678530342864570245312849318820645112295832022960575} a^{14} - \frac{463535372436923561732961240900631048474118898716}{347141213714582809812513972752825804491833280918423} a^{13} + \frac{3116346846495385739273396183820253970640991883962}{8678530342864570245312849318820645112295832022960575} a^{12} - \frac{141034014908249092343414498425221391795804288526381}{8678530342864570245312849318820645112295832022960575} a^{11} + \frac{7899833194072770488135265783027807080654265574759}{1735706068572914049062569863764129022459166404592115} a^{10} + \frac{1954284763384406290678489616257660304524911965240724}{8678530342864570245312849318820645112295832022960575} a^{9} + \frac{47442330222459700656374141481487955735568186523497}{347141213714582809812513972752825804491833280918423} a^{8} + \frac{590049293222741042670337806675082781871291484469801}{1735706068572914049062569863764129022459166404592115} a^{7} - \frac{4097907473555200354644292444021086704951275240828149}{8678530342864570245312849318820645112295832022960575} a^{6} + \frac{194682170138844806008444660348957264533416506623932}{8678530342864570245312849318820645112295832022960575} a^{5} + \frac{2406223135811428197877707008366906737410631867349983}{8678530342864570245312849318820645112295832022960575} a^{4} + \frac{342868619035523401112290421557233510816732340140378}{1735706068572914049062569863764129022459166404592115} a^{3} + \frac{787650177661158772995031281623852099621239091265331}{8678530342864570245312849318820645112295832022960575} a^{2} - \frac{320614682479734885623853888431725876473919687148862}{1735706068572914049062569863764129022459166404592115} a - \frac{2613273424380878365901651784282764165843040435831839}{8678530342864570245312849318820645112295832022960575}$

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

Class group and class number

$C_{10}\times C_{41764040}$, which has order $417640400$ (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:  \( 19344397.966990974 \) (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

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{-85}) \), \(\Q(\sqrt{10}, \sqrt{-34})\), 5.5.390625.1, 10.10.25000000000000000.1, 10.0.7099285000000000000000.1, 10.0.1109263281250000000000.3

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.17.29$x^{10} - 10 x^{8} + 35$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.17.29$x^{10} - 10 x^{8} + 35$$10$$1$$17$$C_{10}$$[2]_{2}$
17Data not computed