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Properties

Label 20.0.65226831883...8049.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 101^{10}$
Root discriminant $17.41$
Ramified primes $3, 101$
Class number $1$
Class group Trivial
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![289, 221, -69, 114, 374, 527, 29, -298, 157, -23, 154, -4, -32, 4, 2, 13, 5, -6, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 3*x^18 - 6*x^17 + 5*x^16 + 13*x^15 + 2*x^14 + 4*x^13 - 32*x^12 - 4*x^11 + 154*x^10 - 23*x^9 + 157*x^8 - 298*x^7 + 29*x^6 + 527*x^5 + 374*x^4 + 114*x^3 - 69*x^2 + 221*x + 289)
 
gp: K = bnfinit(x^20 - 2*x^19 + 3*x^18 - 6*x^17 + 5*x^16 + 13*x^15 + 2*x^14 + 4*x^13 - 32*x^12 - 4*x^11 + 154*x^10 - 23*x^9 + 157*x^8 - 298*x^7 + 29*x^6 + 527*x^5 + 374*x^4 + 114*x^3 - 69*x^2 + 221*x + 289, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 3 x^{18} - 6 x^{17} + 5 x^{16} + 13 x^{15} + 2 x^{14} + 4 x^{13} - 32 x^{12} - 4 x^{11} + 154 x^{10} - 23 x^{9} + 157 x^{8} - 298 x^{7} + 29 x^{6} + 527 x^{5} + 374 x^{4} + 114 x^{3} - 69 x^{2} + 221 x + 289 \)

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:  \(6522683188340621511158049=3^{10}\cdot 101^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.41$
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:  $3, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{45} a^{16} + \frac{1}{45} a^{15} + \frac{1}{45} a^{14} - \frac{1}{9} a^{13} - \frac{1}{45} a^{12} + \frac{4}{45} a^{11} + \frac{16}{45} a^{9} + \frac{17}{45} a^{8} + \frac{11}{45} a^{7} - \frac{1}{3} a^{6} + \frac{14}{45} a^{5} - \frac{16}{45} a^{4} + \frac{4}{9} a^{3} - \frac{14}{45} a^{2} - \frac{4}{45} a + \frac{16}{45}$, $\frac{1}{45} a^{17} - \frac{2}{15} a^{14} + \frac{4}{45} a^{13} + \frac{1}{9} a^{12} - \frac{4}{45} a^{11} + \frac{16}{45} a^{10} + \frac{1}{45} a^{9} - \frac{2}{15} a^{8} + \frac{19}{45} a^{7} - \frac{16}{45} a^{6} + \frac{1}{3} a^{5} - \frac{1}{5} a^{4} + \frac{11}{45} a^{3} + \frac{2}{9} a^{2} + \frac{4}{9} a - \frac{16}{45}$, $\frac{1}{5433435} a^{18} - \frac{6982}{1086687} a^{17} - \frac{16426}{5433435} a^{16} + \frac{199748}{5433435} a^{15} - \frac{295814}{1811145} a^{14} + \frac{66973}{1086687} a^{13} - \frac{435073}{5433435} a^{12} + \frac{243587}{5433435} a^{11} + \frac{736766}{5433435} a^{10} - \frac{804004}{1811145} a^{9} + \frac{375887}{5433435} a^{8} + \frac{2400928}{5433435} a^{7} + \frac{325072}{1086687} a^{6} + \frac{1297117}{5433435} a^{5} + \frac{554899}{1811145} a^{4} + \frac{82934}{1086687} a^{3} + \frac{1231754}{5433435} a^{2} - \frac{1044157}{5433435} a - \frac{1867721}{5433435}$, $\frac{1}{21086257422072646125} a^{19} - \frac{166639959328}{7028752474024215375} a^{18} + \frac{178521927588033866}{21086257422072646125} a^{17} + \frac{28925693206682851}{3012322488867520875} a^{16} - \frac{1419462123277990819}{21086257422072646125} a^{15} + \frac{101579160858875507}{7028752474024215375} a^{14} + \frac{689547223497981601}{4217251484414529225} a^{13} - \frac{907375493434428281}{21086257422072646125} a^{12} + \frac{155722857842804873}{468583498268281025} a^{11} + \frac{6819975357244086251}{21086257422072646125} a^{10} - \frac{2791670686011128}{21086257422072646125} a^{9} + \frac{8418570172380173173}{21086257422072646125} a^{8} + \frac{10371148833699573721}{21086257422072646125} a^{7} - \frac{340082129578931989}{4217251484414529225} a^{6} - \frac{8437641218968160231}{21086257422072646125} a^{5} - \frac{40473172409743652}{137818675961259125} a^{4} + \frac{401050030786599148}{1240368083651332125} a^{3} + \frac{2011284788530884152}{21086257422072646125} a^{2} + \frac{168050944547226952}{1004107496289173625} a + \frac{343656732878799931}{1240368083651332125}$

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:  \( -\frac{19433228716}{135829178001125} a^{19} - \frac{582376012693}{407487534003375} a^{18} + \frac{1596854478707}{407487534003375} a^{17} - \frac{156100842241}{19404168285875} a^{16} + \frac{6889488106112}{407487534003375} a^{15} - \frac{9217727191583}{407487534003375} a^{14} - \frac{356270378998}{81497506800675} a^{13} - \frac{2302704999187}{407487534003375} a^{12} - \frac{538155739272}{27165835600225} a^{11} + \frac{24786995375777}{407487534003375} a^{10} - \frac{17196748524656}{407487534003375} a^{9} - \frac{79764115586129}{407487534003375} a^{8} + \frac{21070046628439}{135829178001125} a^{7} - \frac{14061767208451}{27165835600225} a^{6} + \frac{374827088304538}{407487534003375} a^{5} - \frac{16340035415111}{23969854941375} a^{4} - \frac{5267986694654}{23969854941375} a^{3} - \frac{61145739121307}{135829178001125} a^{2} - \frac{31393687429363}{58212504857625} a + \frac{5047516768479}{7989951647125} \) (order $6$)
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:  \( 56202.6193366 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-303}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{101}) \), \(\Q(\sqrt{-3}, \sqrt{101})\), 5.1.91809.1 x5, 10.0.2553954421743.1, 10.0.25286677443.1 x5, 10.2.851318140581.1 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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$101$101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$