Properties

Label 20.0.54054163264...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{10}\cdot 61^{5}$
Root discriminant $10.88$
Ramified primes $2, 5, 61$
Class number $1$
Class group Trivial
Galois Group $D_5\wr C_2$ (as 20T48)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 7, -5, 4, 11, -9, 21, 18, -9, 49, -9, 18, 21, -9, 11, 4, -5, 7, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 7*x^18 - 5*x^17 + 4*x^16 + 11*x^15 - 9*x^14 + 21*x^13 + 18*x^12 - 9*x^11 + 49*x^10 - 9*x^9 + 18*x^8 + 21*x^7 - 9*x^6 + 11*x^5 + 4*x^4 - 5*x^3 + 7*x^2 - 3*x + 1)
gp: K = bnfinit(x^20 - 3*x^19 + 7*x^18 - 5*x^17 + 4*x^16 + 11*x^15 - 9*x^14 + 21*x^13 + 18*x^12 - 9*x^11 + 49*x^10 - 9*x^9 + 18*x^8 + 21*x^7 - 9*x^6 + 11*x^5 + 4*x^4 - 5*x^3 + 7*x^2 - 3*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 3 x^{19} \) \(\mathstrut +\mathstrut 7 x^{18} \) \(\mathstrut -\mathstrut 5 x^{17} \) \(\mathstrut +\mathstrut 4 x^{16} \) \(\mathstrut +\mathstrut 11 x^{15} \) \(\mathstrut -\mathstrut 9 x^{14} \) \(\mathstrut +\mathstrut 21 x^{13} \) \(\mathstrut +\mathstrut 18 x^{12} \) \(\mathstrut -\mathstrut 9 x^{11} \) \(\mathstrut +\mathstrut 49 x^{10} \) \(\mathstrut -\mathstrut 9 x^{9} \) \(\mathstrut +\mathstrut 18 x^{8} \) \(\mathstrut +\mathstrut 21 x^{7} \) \(\mathstrut -\mathstrut 9 x^{6} \) \(\mathstrut +\mathstrut 11 x^{5} \) \(\mathstrut +\mathstrut 4 x^{4} \) \(\mathstrut -\mathstrut 5 x^{3} \) \(\mathstrut +\mathstrut 7 x^{2} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(540541632640000000000=2^{16}\cdot 5^{10}\cdot 61^{5}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.88$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 5, 61$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{11} a^{17} + \frac{5}{11} a^{16} - \frac{5}{11} a^{15} + \frac{4}{11} a^{14} + \frac{4}{11} a^{13} - \frac{5}{11} a^{12} - \frac{1}{11} a^{10} + \frac{5}{11} a^{9} - \frac{5}{11} a^{8} + \frac{1}{11} a^{7} + \frac{5}{11} a^{5} - \frac{4}{11} a^{4} - \frac{4}{11} a^{3} + \frac{5}{11} a^{2} - \frac{5}{11} a - \frac{1}{11}$, $\frac{1}{121319} a^{18} + \frac{684}{121319} a^{17} - \frac{37420}{121319} a^{16} - \frac{32717}{121319} a^{15} - \frac{39256}{121319} a^{14} - \frac{58471}{121319} a^{13} - \frac{39915}{121319} a^{12} - \frac{1349}{2959} a^{11} + \frac{48584}{121319} a^{10} - \frac{29478}{121319} a^{9} + \frac{26526}{121319} a^{8} + \frac{265}{2959} a^{7} + \frac{48317}{121319} a^{6} + \frac{7703}{121319} a^{5} + \frac{37947}{121319} a^{4} + \frac{33457}{121319} a^{3} - \frac{37420}{121319} a^{2} - \frac{32403}{121319} a + \frac{55146}{121319}$, $\frac{1}{121319} a^{19} + \frac{2058}{121319} a^{17} - \frac{46775}{121319} a^{16} + \frac{27505}{121319} a^{15} - \frac{4723}{11029} a^{14} + \frac{7211}{121319} a^{13} - \frac{39195}{121319} a^{12} + \frac{28412}{121319} a^{11} - \frac{41586}{121319} a^{10} + \frac{39495}{121319} a^{9} - \frac{45359}{121319} a^{8} + \frac{39174}{121319} a^{7} - \frac{42357}{121319} a^{6} - \frac{25217}{121319} a^{5} - \frac{107}{269} a^{4} + \frac{3670}{11029} a^{3} - \frac{46461}{121319} a^{2} + \frac{28450}{121319} a - \frac{11713}{121319}$

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

Class group and class number

Trivial Abelian group, 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:  \( \frac{66041}{121319} a^{19} - \frac{85293}{121319} a^{18} + \frac{170444}{121319} a^{17} + \frac{29488}{11029} a^{16} + \frac{21080}{121319} a^{15} + \frac{944045}{121319} a^{14} + \frac{879360}{121319} a^{13} + \frac{850739}{121319} a^{12} + \frac{3172754}{121319} a^{11} + \frac{2482847}{121319} a^{10} + \frac{3015868}{121319} a^{9} + \frac{4661064}{121319} a^{8} + \frac{2557554}{121319} a^{7} + \frac{2848432}{121319} a^{6} + \frac{2710759}{121319} a^{5} + \frac{650999}{121319} a^{4} + \frac{956995}{121319} a^{3} + \frac{558331}{121319} a^{2} - \frac{19963}{121319} a + \frac{204201}{121319} \),  \( \frac{31585}{121319} a^{19} - \frac{94507}{121319} a^{18} + \frac{204736}{121319} a^{17} - \frac{127938}{121319} a^{16} + \frac{6697}{11029} a^{15} + \frac{288670}{121319} a^{14} - \frac{250329}{121319} a^{13} + \frac{448312}{121319} a^{12} + \frac{418882}{121319} a^{11} - \frac{393081}{121319} a^{10} + \frac{881883}{121319} a^{9} - \frac{648608}{121319} a^{8} + \frac{92002}{121319} a^{7} - \frac{272748}{121319} a^{6} - \frac{67622}{11029} a^{5} - \frac{123208}{121319} a^{4} - \frac{434089}{121319} a^{3} - \frac{279725}{121319} a^{2} + \frac{3656}{121319} a - \frac{18466}{11029} \),  \( \frac{2043}{121319} a^{19} - \frac{1472}{11029} a^{18} + \frac{22291}{121319} a^{17} - \frac{34689}{121319} a^{16} - \frac{3220}{11029} a^{15} - \frac{31175}{121319} a^{14} - \frac{168121}{121319} a^{13} - \frac{99826}{121319} a^{12} - \frac{200934}{121319} a^{11} - \frac{539848}{121319} a^{10} - \frac{424267}{121319} a^{9} - \frac{69036}{11029} a^{8} - \frac{680241}{121319} a^{7} - \frac{482813}{121319} a^{6} - \frac{62335}{11029} a^{5} - \frac{432156}{121319} a^{4} - \frac{221825}{121319} a^{3} - \frac{242420}{121319} a^{2} - \frac{34779}{121319} a - \frac{145269}{121319} \),  \( \frac{96475}{121319} a^{19} - \frac{321367}{121319} a^{18} + \frac{700290}{121319} a^{17} - \frac{538148}{121319} a^{16} + \frac{174885}{121319} a^{15} + \frac{91478}{11029} a^{14} - \frac{1338410}{121319} a^{13} + \frac{1396947}{121319} a^{12} + \frac{1220117}{121319} a^{11} - \frac{2573052}{121319} a^{10} + \frac{2922085}{121319} a^{9} - \frac{2616227}{121319} a^{8} - \frac{1068996}{121319} a^{7} + \frac{247092}{121319} a^{6} - \frac{57249}{2959} a^{5} - \frac{39501}{11029} a^{4} - \frac{10248}{121319} a^{3} - \frac{696067}{121319} a^{2} + \frac{382680}{121319} a - \frac{246557}{121319} \),  \( \frac{19698}{121319} a^{19} - \frac{159072}{121319} a^{18} + \frac{32334}{11029} a^{17} - \frac{583177}{121319} a^{16} + \frac{98259}{121319} a^{15} + \frac{20609}{121319} a^{14} - \frac{1476786}{121319} a^{13} + \frac{360601}{121319} a^{12} - \frac{1381431}{121319} a^{11} - \frac{3337747}{121319} a^{10} - \frac{121751}{121319} a^{9} - \frac{4911017}{121319} a^{8} - \frac{2174052}{121319} a^{7} - \frac{1816425}{121319} a^{6} - \frac{3307751}{121319} a^{5} - \frac{419172}{121319} a^{4} - \frac{760566}{121319} a^{3} - \frac{827912}{121319} a^{2} + \frac{307201}{121319} a - \frac{386675}{121319} \),  \( a \),  \( \frac{8914}{11029} a^{19} - \frac{290874}{121319} a^{18} + \frac{620487}{121319} a^{17} - \frac{290771}{121319} a^{16} - \frac{3643}{121319} a^{15} + \frac{1314292}{121319} a^{14} - \frac{856956}{121319} a^{13} + \frac{1302787}{121319} a^{12} + \frac{2447226}{121319} a^{11} - \frac{1543496}{121319} a^{10} + \frac{3535194}{121319} a^{9} + \frac{495903}{121319} a^{8} - \frac{175248}{121319} a^{7} + \frac{2306926}{121319} a^{6} - \frac{395063}{121319} a^{5} + \frac{73688}{121319} a^{4} + \frac{857396}{121319} a^{3} - \frac{316841}{121319} a^{2} + \frac{231080}{121319} a - \frac{49823}{121319} \),  \( \frac{125545}{121319} a^{19} - \frac{383902}{121319} a^{18} + \frac{833924}{121319} a^{17} - \frac{518305}{121319} a^{16} + \frac{198772}{121319} a^{15} + \frac{1402789}{121319} a^{14} - \frac{116470}{11029} a^{13} + \frac{1890808}{121319} a^{12} + \frac{204791}{11029} a^{11} - \frac{204107}{11029} a^{10} + \frac{4387304}{121319} a^{9} - \frac{1591230}{121319} a^{8} - \frac{243237}{121319} a^{7} + \frac{1597561}{121319} a^{6} - \frac{2014487}{121319} a^{5} - \frac{128010}{121319} a^{4} + \frac{524162}{121319} a^{3} - \frac{889807}{121319} a^{2} + \frac{598630}{121319} a - \frac{208424}{121319} \),  \( \frac{47071}{121319} a^{19} - \frac{200235}{121319} a^{18} + \frac{476040}{121319} a^{17} - \frac{545206}{121319} a^{16} + \frac{219997}{121319} a^{15} + \frac{533672}{121319} a^{14} - \frac{1250355}{121319} a^{13} + \frac{1069741}{121319} a^{12} + \frac{13936}{11029} a^{11} - \frac{2372305}{121319} a^{10} + \frac{2253815}{121319} a^{9} - \frac{2613839}{121319} a^{8} - \frac{834211}{121319} a^{7} + \frac{405854}{121319} a^{6} - \frac{1927912}{121319} a^{5} - \frac{213330}{121319} a^{4} + \frac{75732}{121319} a^{3} - \frac{808468}{121319} a^{2} + \frac{268790}{121319} a - \frac{181090}{121319} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 75.025557452 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_5\wr C_2$ (as 20T48):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 200
The 14 conjugacy class representatives for $D_5\wr C_2$
Character table for $D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.1525.1, 10.2.2976800000.1 x5

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 25 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$