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Properties

Label 20.0.11705855471...8881.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{18}\cdot 1451^{2}$
Root discriminant $17.92$
Ramified primes $11, 1451$
Class number $1$
Class group Trivial
Galois group $C_2^2\times C_2^4:C_5$ (as 20T86)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121, -605, 726, 1089, -3025, 22, 7403, -11957, 10494, -7216, 5369, -4196, 2918, -1852, 1098, -562, 247, -102, 37, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 37*x^18 - 102*x^17 + 247*x^16 - 562*x^15 + 1098*x^14 - 1852*x^13 + 2918*x^12 - 4196*x^11 + 5369*x^10 - 7216*x^9 + 10494*x^8 - 11957*x^7 + 7403*x^6 + 22*x^5 - 3025*x^4 + 1089*x^3 + 726*x^2 - 605*x + 121)
 
gp: K = bnfinit(x^20 - 9*x^19 + 37*x^18 - 102*x^17 + 247*x^16 - 562*x^15 + 1098*x^14 - 1852*x^13 + 2918*x^12 - 4196*x^11 + 5369*x^10 - 7216*x^9 + 10494*x^8 - 11957*x^7 + 7403*x^6 + 22*x^5 - 3025*x^4 + 1089*x^3 + 726*x^2 - 605*x + 121, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 37 x^{18} - 102 x^{17} + 247 x^{16} - 562 x^{15} + 1098 x^{14} - 1852 x^{13} + 2918 x^{12} - 4196 x^{11} + 5369 x^{10} - 7216 x^{9} + 10494 x^{8} - 11957 x^{7} + 7403 x^{6} + 22 x^{5} - 3025 x^{4} + 1089 x^{3} + 726 x^{2} - 605 x + 121 \)

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:  \(11705855471743857652328881=11^{18}\cdot 1451^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.92$
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:  $11, 1451$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{33} a^{15} + \frac{13}{33} a^{14} + \frac{5}{11} a^{13} + \frac{8}{33} a^{12} + \frac{16}{33} a^{11} + \frac{10}{33} a^{10} + \frac{3}{11} a^{9} + \frac{7}{33} a^{8} + \frac{14}{33} a^{7} + \frac{2}{11} a^{6} + \frac{1}{33} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{33} a^{16} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{2}{33} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{33} a^{17} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{2}{33} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{1419} a^{18} + \frac{4}{473} a^{17} - \frac{10}{1419} a^{16} + \frac{13}{1419} a^{15} + \frac{70}{1419} a^{14} - \frac{133}{473} a^{13} + \frac{42}{473} a^{12} - \frac{202}{473} a^{11} - \frac{118}{473} a^{10} - \frac{587}{1419} a^{9} - \frac{219}{473} a^{8} - \frac{601}{1419} a^{7} - \frac{158}{473} a^{6} + \frac{41}{129} a^{5} - \frac{22}{129} a^{4} - \frac{34}{129} a^{3} - \frac{1}{129} a^{2} + \frac{10}{129} a + \frac{16}{129}$, $\frac{1}{917025810244802709} a^{19} - \frac{177921434883155}{917025810244802709} a^{18} - \frac{10982198461689094}{917025810244802709} a^{17} - \frac{139439825998442}{27788660916509173} a^{16} - \frac{3217530151540130}{305675270081600903} a^{15} - \frac{283389067925488573}{917025810244802709} a^{14} - \frac{16664245718297055}{305675270081600903} a^{13} - \frac{190446178003174087}{917025810244802709} a^{12} + \frac{397901366715832849}{917025810244802709} a^{11} + \frac{283889672873856854}{917025810244802709} a^{10} - \frac{212121346356008129}{917025810244802709} a^{9} - \frac{73191860669398216}{305675270081600903} a^{8} - \frac{398058738938752202}{917025810244802709} a^{7} - \frac{147285069389298101}{917025810244802709} a^{6} + \frac{99224185744845087}{305675270081600903} a^{5} - \frac{28033972873825216}{83365982749527519} a^{4} - \frac{29660695649005918}{83365982749527519} a^{3} + \frac{23806901802600314}{83365982749527519} a^{2} - \frac{28498807413542516}{83365982749527519} a - \frac{10194401572212719}{27788660916509173}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{123572754562405451}{305675270081600903} a^{19} + \frac{3053726851997467814}{917025810244802709} a^{18} - \frac{11407950702588984073}{917025810244802709} a^{17} + \frac{29263442814103862098}{917025810244802709} a^{16} - \frac{69750873428080209043}{917025810244802709} a^{15} + \frac{156341738437999986647}{917025810244802709} a^{14} - \frac{290665401819069232295}{917025810244802709} a^{13} + \frac{471149312360882065693}{917025810244802709} a^{12} - \frac{244435331588338852330}{305675270081600903} a^{11} + \frac{1013352113552741132992}{917025810244802709} a^{10} - \frac{1244099890905519258056}{917025810244802709} a^{9} + \frac{1760727251372928824251}{917025810244802709} a^{8} - \frac{2588469435211128045596}{917025810244802709} a^{7} + \frac{229361057593751860648}{83365982749527519} a^{6} - \frac{911410759635637385447}{917025810244802709} a^{5} - \frac{57924441639799081162}{83365982749527519} a^{4} + \frac{19276248769879161755}{27788660916509173} a^{3} + \frac{4742458042331526455}{83365982749527519} a^{2} - \frac{20201043422844773818}{83365982749527519} a + \frac{1928051947934108454}{27788660916509173} \) (order $22$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 177120.841849 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T86):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{11})^+\), 10.10.3421382099641.1, \(\Q(\zeta_{11})\), 10.0.311034736331.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
1451Data not computed