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Properties

Label 20.0.97756416090...9552.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 3^{10}\cdot 11^{16}$
Root discriminant $79.34$
Ramified primes $2, 3, 11$
Class number $131042$ (GRH)
Class group $[131042]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![95829823, -47685756, 95586810, -38829056, 47163703, -21370648, 17951094, -5786136, 4088499, -1402508, 886308, -282888, 149216, -38000, 16682, -3204, 1183, -160, 50, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 50*x^18 - 160*x^17 + 1183*x^16 - 3204*x^15 + 16682*x^14 - 38000*x^13 + 149216*x^12 - 282888*x^11 + 886308*x^10 - 1402508*x^9 + 4088499*x^8 - 5786136*x^7 + 17951094*x^6 - 21370648*x^5 + 47163703*x^4 - 38829056*x^3 + 95586810*x^2 - 47685756*x + 95829823)
 
gp: K = bnfinit(x^20 - 4*x^19 + 50*x^18 - 160*x^17 + 1183*x^16 - 3204*x^15 + 16682*x^14 - 38000*x^13 + 149216*x^12 - 282888*x^11 + 886308*x^10 - 1402508*x^9 + 4088499*x^8 - 5786136*x^7 + 17951094*x^6 - 21370648*x^5 + 47163703*x^4 - 38829056*x^3 + 95586810*x^2 - 47685756*x + 95829823, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 50 x^{18} - 160 x^{17} + 1183 x^{16} - 3204 x^{15} + 16682 x^{14} - 38000 x^{13} + 149216 x^{12} - 282888 x^{11} + 886308 x^{10} - 1402508 x^{9} + 4088499 x^{8} - 5786136 x^{7} + 17951094 x^{6} - 21370648 x^{5} + 47163703 x^{4} - 38829056 x^{3} + 95586810 x^{2} - 47685756 x + 95829823 \)

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:  \(97756416090547441671665711234313879552=2^{55}\cdot 3^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.34$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(528=2^{4}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{528}(1,·)$, $\chi_{528}(389,·)$, $\chi_{528}(289,·)$, $\chi_{528}(265,·)$, $\chi_{528}(269,·)$, $\chi_{528}(25,·)$, $\chi_{528}(221,·)$, $\chi_{528}(5,·)$, $\chi_{528}(97,·)$, $\chi_{528}(485,·)$, $\chi_{528}(433,·)$, $\chi_{528}(169,·)$, $\chi_{528}(317,·)$, $\chi_{528}(49,·)$, $\chi_{528}(245,·)$, $\chi_{528}(361,·)$, $\chi_{528}(313,·)$, $\chi_{528}(509,·)$, $\chi_{528}(125,·)$, $\chi_{528}(53,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{69} a^{15} + \frac{5}{69} a^{14} - \frac{3}{23} a^{13} + \frac{7}{69} a^{12} + \frac{5}{69} a^{10} - \frac{11}{23} a^{9} + \frac{25}{69} a^{8} + \frac{32}{69} a^{7} - \frac{8}{23} a^{6} + \frac{28}{69} a^{5} + \frac{34}{69} a^{4} - \frac{5}{23} a^{3} - \frac{4}{23} a^{2} - \frac{9}{23} a + \frac{2}{23}$, $\frac{1}{69} a^{16} - \frac{11}{69} a^{14} + \frac{2}{23} a^{13} + \frac{11}{69} a^{12} + \frac{5}{69} a^{11} + \frac{11}{69} a^{10} + \frac{2}{23} a^{9} - \frac{8}{23} a^{8} - \frac{13}{69} a^{6} + \frac{32}{69} a^{5} - \frac{1}{69} a^{4} - \frac{29}{69} a^{3} + \frac{11}{23} a^{2} + \frac{26}{69} a - \frac{7}{69}$, $\frac{1}{69} a^{17} - \frac{8}{69} a^{14} + \frac{4}{69} a^{13} - \frac{10}{69} a^{12} + \frac{11}{69} a^{11} - \frac{8}{69} a^{10} + \frac{4}{69} a^{9} - \frac{1}{69} a^{8} + \frac{17}{69} a^{7} - \frac{25}{69} a^{6} - \frac{5}{23} a^{5} + \frac{1}{3} a^{4} - \frac{17}{69} a^{3} - \frac{14}{69} a^{2} - \frac{28}{69} a - \frac{26}{69}$, $\frac{1}{3178552014864600173781} a^{18} + \frac{765261677376535858}{1059517338288200057927} a^{17} + \frac{21864375385911364604}{3178552014864600173781} a^{16} - \frac{4001737349557489076}{3178552014864600173781} a^{15} - \frac{5836847295227750355}{46065971229921741649} a^{14} - \frac{264961818760054719767}{3178552014864600173781} a^{13} - \frac{22930222279628117539}{3178552014864600173781} a^{12} + \frac{231019599213776385334}{3178552014864600173781} a^{11} + \frac{461881308084115900684}{3178552014864600173781} a^{10} + \frac{99995753829212711414}{1059517338288200057927} a^{9} - \frac{275993971640834901923}{1059517338288200057927} a^{8} + \frac{940595832482715714130}{3178552014864600173781} a^{7} + \frac{151092254799246820522}{1059517338288200057927} a^{6} - \frac{1321979006878177089451}{3178552014864600173781} a^{5} - \frac{915418303539398776235}{3178552014864600173781} a^{4} + \frac{571682941614506702228}{3178552014864600173781} a^{3} - \frac{800531789327486297602}{3178552014864600173781} a^{2} - \frac{164242170356140065219}{1059517338288200057927} a - \frac{1046670850825342286678}{3178552014864600173781}$, $\frac{1}{108159701222931031384546881143930082846411811107} a^{19} - \frac{12452689409263607819588896}{108159701222931031384546881143930082846411811107} a^{18} + \frac{392318452576164603623648751418339942286102555}{108159701222931031384546881143930082846411811107} a^{17} - \frac{130040175232704124570280790417264601891260418}{36053233740977010461515627047976694282137270369} a^{16} + \frac{159345098677980207719156486946535506364305325}{36053233740977010461515627047976694282137270369} a^{15} - \frac{17109974989891792506043254352574526556219549160}{108159701222931031384546881143930082846411811107} a^{14} + \frac{1359331584946886297570453519086524326451708838}{108159701222931031384546881143930082846411811107} a^{13} - \frac{13659124535074021456603693940050243264698605321}{108159701222931031384546881143930082846411811107} a^{12} + \frac{1411499800454362943378284414831025861426788768}{108159701222931031384546881143930082846411811107} a^{11} + \frac{2531245512017318854392750833901503446511988652}{108159701222931031384546881143930082846411811107} a^{10} + \frac{1531468885715854766195115242673176997164898184}{108159701222931031384546881143930082846411811107} a^{9} + \frac{53776839722491975093159754298245768371123766272}{108159701222931031384546881143930082846411811107} a^{8} + \frac{4210087827229703468918773177503195090358964505}{36053233740977010461515627047976694282137270369} a^{7} - \frac{22546171104613696323673275611777674085549322715}{108159701222931031384546881143930082846411811107} a^{6} + \frac{12941115241137417836853440749852011450765911592}{36053233740977010461515627047976694282137270369} a^{5} + \frac{41961881450904604901943094378262540425001038549}{108159701222931031384546881143930082846411811107} a^{4} + \frac{20523294599182852456557105791424809753635606720}{108159701222931031384546881143930082846411811107} a^{3} + \frac{2920826303291420456834927104968459067180209343}{36053233740977010461515627047976694282137270369} a^{2} + \frac{13075613846468819623892975086293948904176126069}{36053233740977010461515627047976694282137270369} a + \frac{50653089097270264890459518158541934005342030908}{108159701222931031384546881143930082846411811107}$

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

Class group and class number

$C_{131042}$, which has order $131042$ (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:  \( 530208.250733 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

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

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.18432.2, \(\Q(\zeta_{11})^+\), 10.10.7024111812608.1

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 R R $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ $20$

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
2Data not computed
3Data not computed
11Data not computed