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Properties

Label 20.0.20368198195...0961.2
Degree $20$
Signature $[0, 10]$
Discriminant $11^{16}\cdot 1451^{4}$
Root discriminant $29.20$
Ramified primes $11, 1451$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^4:C_5$ (as 20T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2837, 1831, -3846, 8834, -5227, -11082, 45374, -83184, 112680, -111873, 93279, -63382, 37604, -18705, 8238, -3008, 972, -247, 55, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 55*x^18 - 247*x^17 + 972*x^16 - 3008*x^15 + 8238*x^14 - 18705*x^13 + 37604*x^12 - 63382*x^11 + 93279*x^10 - 111873*x^9 + 112680*x^8 - 83184*x^7 + 45374*x^6 - 11082*x^5 - 5227*x^4 + 8834*x^3 - 3846*x^2 + 1831*x + 2837)
 
gp: K = bnfinit(x^20 - 8*x^19 + 55*x^18 - 247*x^17 + 972*x^16 - 3008*x^15 + 8238*x^14 - 18705*x^13 + 37604*x^12 - 63382*x^11 + 93279*x^10 - 111873*x^9 + 112680*x^8 - 83184*x^7 + 45374*x^6 - 11082*x^5 - 5227*x^4 + 8834*x^3 - 3846*x^2 + 1831*x + 2837, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 55 x^{18} - 247 x^{17} + 972 x^{16} - 3008 x^{15} + 8238 x^{14} - 18705 x^{13} + 37604 x^{12} - 63382 x^{11} + 93279 x^{10} - 111873 x^{9} + 112680 x^{8} - 83184 x^{7} + 45374 x^{6} - 11082 x^{5} - 5227 x^{4} + 8834 x^{3} - 3846 x^{2} + 1831 x + 2837 \)

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:  \(203681981950950327645213870961=11^{16}\cdot 1451^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.20$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{33} a^{15} - \frac{2}{11} a^{14} + \frac{5}{11} a^{13} + \frac{5}{11} a^{12} - \frac{4}{33} a^{11} - \frac{1}{3} a^{10} - \frac{2}{11} a^{9} + \frac{4}{33} a^{8} - \frac{5}{11} a^{7} - \frac{5}{33} a^{6} - \frac{4}{33} a^{5} - \frac{16}{33} a^{4} - \frac{1}{33} a^{3} + \frac{1}{11} a^{2} - \frac{13}{33} a - \frac{1}{33}$, $\frac{1}{33} a^{16} + \frac{4}{11} a^{14} + \frac{2}{11} a^{13} - \frac{13}{33} a^{12} - \frac{2}{33} a^{11} - \frac{2}{11} a^{10} + \frac{1}{33} a^{9} + \frac{3}{11} a^{8} + \frac{4}{33} a^{7} - \frac{1}{33} a^{6} - \frac{7}{33} a^{5} + \frac{2}{33} a^{4} - \frac{1}{11} a^{3} + \frac{5}{33} a^{2} - \frac{13}{33} a - \frac{2}{11}$, $\frac{1}{33} a^{17} + \frac{4}{11} a^{14} + \frac{5}{33} a^{13} + \frac{16}{33} a^{12} + \frac{3}{11} a^{11} + \frac{1}{33} a^{10} + \frac{5}{11} a^{9} - \frac{1}{3} a^{8} + \frac{14}{33} a^{7} - \frac{13}{33} a^{6} - \frac{16}{33} a^{5} - \frac{3}{11} a^{4} - \frac{16}{33} a^{3} - \frac{16}{33} a^{2} - \frac{5}{11} a + \frac{4}{11}$, $\frac{1}{3597} a^{18} - \frac{14}{3597} a^{17} - \frac{10}{1199} a^{16} + \frac{10}{3597} a^{15} + \frac{314}{3597} a^{14} - \frac{31}{109} a^{13} + \frac{1531}{3597} a^{12} - \frac{173}{1199} a^{11} - \frac{457}{3597} a^{10} - \frac{833}{3597} a^{9} - \frac{127}{327} a^{8} - \frac{101}{3597} a^{7} + \frac{470}{3597} a^{6} + \frac{1588}{3597} a^{5} + \frac{16}{3597} a^{4} + \frac{375}{1199} a^{3} - \frac{838}{3597} a^{2} + \frac{118}{327} a + \frac{278}{3597}$, $\frac{1}{205988749274639868804484808996725221} a^{19} + \frac{20016354126104643694068379773125}{205988749274639868804484808996725221} a^{18} + \frac{1572571204517562421230007774722676}{205988749274639868804484808996725221} a^{17} - \frac{132443350966710951643684373164825}{68662916424879956268161602998908407} a^{16} - \frac{998309989120056011937639787026174}{68662916424879956268161602998908407} a^{15} + \frac{41929686838247997005839519212172367}{205988749274639868804484808996725221} a^{14} + \frac{20623479801761469927012911701519237}{205988749274639868804484808996725221} a^{13} - \frac{8826669349589898182298646208925809}{18726249934058169891316800817884111} a^{12} - \frac{98970455537075855260367627445615839}{205988749274639868804484808996725221} a^{11} + \frac{8252942278122528726351224236316470}{68662916424879956268161602998908407} a^{10} + \frac{100929413833823607849184243910510299}{205988749274639868804484808996725221} a^{9} + \frac{58398238514023793880730562919217718}{205988749274639868804484808996725221} a^{8} + \frac{100610549109009201616201541013281525}{205988749274639868804484808996725221} a^{7} + \frac{24115390872658490130932704230130876}{205988749274639868804484808996725221} a^{6} + \frac{32805808721954558260050460169162633}{68662916424879956268161602998908407} a^{5} - \frac{7958961464147284699997623427100110}{18726249934058169891316800817884111} a^{4} + \frac{79144537150517453850737344496518088}{205988749274639868804484808996725221} a^{3} - \frac{26588175469925860630043568031378}{1889805039216879530316374394465369} a^{2} - \frac{64854396013011258883045232205623065}{205988749274639868804484808996725221} a - \frac{69386131593311193731582807233245577}{205988749274639868804484808996725221}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2227699.13388 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 16 conjugacy class representatives for $C_2\times C_2^4:C_5$
Character table for $C_2\times C_2^4:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.0.311034736331.1 x2, 10.10.451311402416281.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
1451Data not computed