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Properties

Label 20.20.5338428090...4801.1
Degree $20$
Signature $[20, 0]$
Discriminant $7^{12}\cdot 199^{12}$
Root discriminant $76.98$
Ramified primes $7, 199$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4:D_5$ (as 20T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7067, -144650, 632356, -115897, -3759592, 5673768, 1117774, -6301742, 1308453, 2903891, -910055, -704949, 233926, 95007, -30388, -7034, 2099, 266, -73, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 73*x^18 + 266*x^17 + 2099*x^16 - 7034*x^15 - 30388*x^14 + 95007*x^13 + 233926*x^12 - 704949*x^11 - 910055*x^10 + 2903891*x^9 + 1308453*x^8 - 6301742*x^7 + 1117774*x^6 + 5673768*x^5 - 3759592*x^4 - 115897*x^3 + 632356*x^2 - 144650*x + 7067)
 
gp: K = bnfinit(x^20 - 4*x^19 - 73*x^18 + 266*x^17 + 2099*x^16 - 7034*x^15 - 30388*x^14 + 95007*x^13 + 233926*x^12 - 704949*x^11 - 910055*x^10 + 2903891*x^9 + 1308453*x^8 - 6301742*x^7 + 1117774*x^6 + 5673768*x^5 - 3759592*x^4 - 115897*x^3 + 632356*x^2 - 144650*x + 7067, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 73 x^{18} + 266 x^{17} + 2099 x^{16} - 7034 x^{15} - 30388 x^{14} + 95007 x^{13} + 233926 x^{12} - 704949 x^{11} - 910055 x^{10} + 2903891 x^{9} + 1308453 x^{8} - 6301742 x^{7} + 1117774 x^{6} + 5673768 x^{5} - 3759592 x^{4} - 115897 x^{3} + 632356 x^{2} - 144650 x + 7067 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(53384280905783887100407820071202644801=7^{12}\cdot 199^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.98$
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:  $7, 199$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{16} + \frac{1}{4} a^{15} + \frac{1}{4} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} + \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{56} a^{18} - \frac{3}{28} a^{17} + \frac{3}{28} a^{16} + \frac{5}{14} a^{15} - \frac{3}{56} a^{14} + \frac{3}{7} a^{13} - \frac{5}{56} a^{12} + \frac{25}{56} a^{11} + \frac{3}{8} a^{10} - \frac{3}{14} a^{9} - \frac{3}{7} a^{8} - \frac{17}{56} a^{7} + \frac{1}{8} a^{6} + \frac{17}{56} a^{5} + \frac{1}{56} a^{4} - \frac{23}{56} a^{3} + \frac{25}{56} a^{2} - \frac{1}{56}$, $\frac{1}{3269741858164364577975529596349501149505374699881456} a^{19} + \frac{4965804003923046673831498243009358703512253579311}{3269741858164364577975529596349501149505374699881456} a^{18} + \frac{42013122663497467685165239433822456502052566968411}{817435464541091144493882399087375287376343674970364} a^{17} - \frac{190657588277190734398368094240402939140319667033669}{1634870929082182288987764798174750574752687349940728} a^{16} + \frac{1011978052046107992420083553989583898274813099525357}{3269741858164364577975529596349501149505374699881456} a^{15} + \frac{92026509766439249539197569364199481051635822234517}{3269741858164364577975529596349501149505374699881456} a^{14} - \frac{560433999473684811208050201888858369194687889898789}{3269741858164364577975529596349501149505374699881456} a^{13} - \frac{71041805195446319614815605585936248849042738071257}{204358866135272786123470599771843821844085918742591} a^{12} + \frac{238677682758712851527562700291687473363498074418887}{1634870929082182288987764798174750574752687349940728} a^{11} + \frac{1175632615447743293136288313416389705539487235778981}{3269741858164364577975529596349501149505374699881456} a^{10} - \frac{33644837285056060753641896820468930022188544278309}{204358866135272786123470599771843821844085918742591} a^{9} + \frac{245296564074762303019477786532729513123726753450603}{3269741858164364577975529596349501149505374699881456} a^{8} - \frac{767571016141398100390629387313466286038015318953425}{1634870929082182288987764798174750574752687349940728} a^{7} + \frac{71923201759986022735480157105323406046269951323383}{817435464541091144493882399087375287376343674970364} a^{6} - \frac{739549427031266874662432905301148546621549410664939}{1634870929082182288987764798174750574752687349940728} a^{5} - \frac{688415085802871165908074402240388948913175409119013}{1634870929082182288987764798174750574752687349940728} a^{4} + \frac{605553435375496752262555039424075894862919628675057}{1634870929082182288987764798174750574752687349940728} a^{3} + \frac{1298706369447453472163035649079292624688618881768221}{3269741858164364577975529596349501149505374699881456} a^{2} + \frac{747943978389787477976649191174405227107321091675811}{3269741858164364577975529596349501149505374699881456} a + \frac{1404468795954948859585180428223871044268417328112567}{3269741858164364577975529596349501149505374699881456}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $19$
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:  \( 4352158906050 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:D_5$ (as 20T45):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 10 conjugacy class representatives for $C_2^4:D_5$
Character table for $C_2^4:D_5$

Intermediate fields

5.5.1940449.1, 10.10.5245121853990193.1, 10.10.5245121853990193.2, 10.10.7306454742608338849.1

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 16 sibling: 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
199Data not computed