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Properties

Label 20.0.22392111040...9121.1
Degree $20$
Signature $[0, 10]$
Discriminant $17^{4}\cdot 401^{9}$
Root discriminant $26.15$
Ramified primes $17, 401$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^4:D_5$ (as 20T87)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 216, 468, -243, -269, -1162, 1410, -1023, 1395, -1183, 487, -348, 474, -357, 324, -231, 111, -54, 18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 18*x^18 - 54*x^17 + 111*x^16 - 231*x^15 + 324*x^14 - 357*x^13 + 474*x^12 - 348*x^11 + 487*x^10 - 1183*x^9 + 1395*x^8 - 1023*x^7 + 1410*x^6 - 1162*x^5 - 269*x^4 - 243*x^3 + 468*x^2 + 216*x + 81)
 
gp: K = bnfinit(x^20 - 4*x^19 + 18*x^18 - 54*x^17 + 111*x^16 - 231*x^15 + 324*x^14 - 357*x^13 + 474*x^12 - 348*x^11 + 487*x^10 - 1183*x^9 + 1395*x^8 - 1023*x^7 + 1410*x^6 - 1162*x^5 - 269*x^4 - 243*x^3 + 468*x^2 + 216*x + 81, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 18 x^{18} - 54 x^{17} + 111 x^{16} - 231 x^{15} + 324 x^{14} - 357 x^{13} + 474 x^{12} - 348 x^{11} + 487 x^{10} - 1183 x^{9} + 1395 x^{8} - 1023 x^{7} + 1410 x^{6} - 1162 x^{5} - 269 x^{4} - 243 x^{3} + 468 x^{2} + 216 x + 81 \)

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:  \(22392111040671733867877719121=17^{4}\cdot 401^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.15$
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:  $17, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3411} a^{18} + \frac{290}{3411} a^{17} - \frac{248}{1137} a^{16} + \frac{132}{379} a^{15} + \frac{412}{1137} a^{14} + \frac{337}{1137} a^{13} - \frac{169}{379} a^{12} + \frac{445}{1137} a^{11} + \frac{341}{1137} a^{10} - \frac{329}{1137} a^{9} + \frac{127}{3411} a^{8} - \frac{853}{3411} a^{7} + \frac{85}{1137} a^{6} + \frac{550}{1137} a^{5} - \frac{244}{1137} a^{4} + \frac{755}{3411} a^{3} + \frac{1030}{3411} a^{2} + \frac{413}{1137} a + \frac{137}{379}$, $\frac{1}{28389291752786936888339114607} a^{19} + \frac{972267058131378152033399}{28389291752786936888339114607} a^{18} - \frac{160444248578772176539725244}{3154365750309659654259901623} a^{17} + \frac{693312016570734428422321504}{3154365750309659654259901623} a^{16} - \frac{3586539656165321515639496579}{9463097250928978962779704869} a^{15} + \frac{4338247489396195898310356992}{9463097250928978962779704869} a^{14} + \frac{33144125278548495800260451}{1051455250103219884753300541} a^{13} - \frac{602388990681231103050383912}{9463097250928978962779704869} a^{12} - \frac{958944917040187662762715921}{9463097250928978962779704869} a^{11} + \frac{2964519183507740268760532440}{9463097250928978962779704869} a^{10} - \frac{11543924705803163937958330943}{28389291752786936888339114607} a^{9} + \frac{2008296362101625952286414973}{28389291752786936888339114607} a^{8} - \frac{1036724714551885689531886867}{3154365750309659654259901623} a^{7} - \frac{3178828624122047920009569539}{9463097250928978962779704869} a^{6} - \frac{3402775323931453161209572615}{9463097250928978962779704869} a^{5} + \frac{7658644750308388156077669959}{28389291752786936888339114607} a^{4} + \frac{905742138076811484512396995}{28389291752786936888339114607} a^{3} - \frac{442779224476311893153432077}{1051455250103219884753300541} a^{2} + \frac{348053493218829316430835443}{1051455250103219884753300541} a - \frac{430197381309677473968650016}{1051455250103219884753300541}$

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:  \( 1639379.50525 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T87):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

5.5.160801.1, 10.6.7472661902689.2

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 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
401Data not computed