Properties

Label 20.0.31059261593...3401.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 47^{10}$
Root discriminant $11.87$
Ramified primes $3, 47$
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![1, -1, -5, 0, 22, 1, -38, -11, 46, 11, -51, 11, 46, -11, -38, 1, 22, 0, -5, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 5*x^18 + 22*x^16 + x^15 - 38*x^14 - 11*x^13 + 46*x^12 + 11*x^11 - 51*x^10 + 11*x^9 + 46*x^8 - 11*x^7 - 38*x^6 + x^5 + 22*x^4 - 5*x^2 - x + 1)
gp: K = bnfinit(x^20 - x^19 - 5*x^18 + 22*x^16 + x^15 - 38*x^14 - 11*x^13 + 46*x^12 + 11*x^11 - 51*x^10 + 11*x^9 + 46*x^8 - 11*x^7 - 38*x^6 + x^5 + 22*x^4 - 5*x^2 - x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut x^{19} \) \(\mathstrut -\mathstrut 5 x^{18} \) \(\mathstrut +\mathstrut 22 x^{16} \) \(\mathstrut +\mathstrut x^{15} \) \(\mathstrut -\mathstrut 38 x^{14} \) \(\mathstrut -\mathstrut 11 x^{13} \) \(\mathstrut +\mathstrut 46 x^{12} \) \(\mathstrut +\mathstrut 11 x^{11} \) \(\mathstrut -\mathstrut 51 x^{10} \) \(\mathstrut +\mathstrut 11 x^{9} \) \(\mathstrut +\mathstrut 46 x^{8} \) \(\mathstrut -\mathstrut 11 x^{7} \) \(\mathstrut -\mathstrut 38 x^{6} \) \(\mathstrut +\mathstrut x^{5} \) \(\mathstrut +\mathstrut 22 x^{4} \) \(\mathstrut -\mathstrut 5 x^{2} \) \(\mathstrut -\mathstrut x \) \(\mathstrut +\mathstrut 1 \)

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:  \(3105926159393528563401=3^{10}\cdot 47^{10}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.87$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 47$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{14} - \frac{1}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{17} - \frac{2}{5} a^{15} - \frac{1}{5} a^{14} - \frac{1}{5} a^{13} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{158455} a^{18} - \frac{7114}{158455} a^{17} - \frac{8651}{158455} a^{16} + \frac{61137}{158455} a^{15} + \frac{68476}{158455} a^{14} + \frac{23128}{158455} a^{13} - \frac{38306}{158455} a^{12} + \frac{5754}{14405} a^{11} - \frac{243}{31691} a^{10} - \frac{72631}{158455} a^{9} - \frac{243}{31691} a^{8} + \frac{5754}{14405} a^{7} - \frac{38306}{158455} a^{6} + \frac{23128}{158455} a^{5} + \frac{68476}{158455} a^{4} + \frac{61137}{158455} a^{3} - \frac{8651}{158455} a^{2} - \frac{7114}{158455} a + \frac{1}{158455}$, $\frac{1}{158455} a^{19} - \frac{1424}{31691} a^{17} - \frac{1537}{158455} a^{16} + \frac{13962}{31691} a^{15} + \frac{1468}{31691} a^{14} - \frac{4126}{14405} a^{13} - \frac{12289}{31691} a^{12} - \frac{56809}{158455} a^{11} - \frac{64498}{158455} a^{10} - \frac{6497}{14405} a^{9} + \frac{71427}{158455} a^{8} + \frac{12911}{31691} a^{7} + \frac{56844}{158455} a^{6} + \frac{61396}{158455} a^{5} + \frac{45349}{158455} a^{4} - \frac{7317}{158455} a^{3} - \frac{69788}{158455} a^{2} + \frac{1532}{158455} a + \frac{7114}{158455}$

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:  \( -\frac{32919}{31691} a^{19} + \frac{45216}{158455} a^{18} + \frac{867594}{158455} a^{17} + \frac{625479}{158455} a^{16} - \frac{3248754}{158455} a^{15} - \frac{230742}{14405} a^{14} + \frac{943475}{31691} a^{13} + \frac{5405924}{158455} a^{12} - \frac{4343186}{158455} a^{11} - \frac{5238968}{158455} a^{10} + \frac{5266152}{158455} a^{9} + \frac{37736}{2365} a^{8} - \frac{99718}{2365} a^{7} - \frac{3087386}{158455} a^{6} + \frac{984228}{31691} a^{5} + \frac{3353628}{158455} a^{4} - \frac{1884874}{158455} a^{3} - \frac{149566}{14405} a^{2} + \frac{192919}{158455} a + \frac{415746}{158455} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{79389}{158455} a^{19} - \frac{61071}{158455} a^{18} - \frac{382121}{158455} a^{17} - \frac{101056}{158455} a^{16} + \frac{1585998}{158455} a^{15} + \frac{376317}{158455} a^{14} - \frac{2431643}{158455} a^{13} - \frac{240940}{31691} a^{12} + \frac{542419}{31691} a^{11} + \frac{1195113}{158455} a^{10} - \frac{3237738}{158455} a^{9} + \frac{281944}{158455} a^{8} + \frac{558145}{31691} a^{7} + \frac{104477}{158455} a^{6} - \frac{2267799}{158455} a^{5} - \frac{504786}{158455} a^{4} + \frac{1378559}{158455} a^{3} + \frac{269322}{158455} a^{2} - \frac{94008}{158455} a - \frac{306564}{158455} \),  \( \frac{121917}{158455} a^{19} - \frac{2443}{31691} a^{18} - \frac{697273}{158455} a^{17} - \frac{522122}{158455} a^{16} + \frac{2509286}{158455} a^{15} + \frac{2311679}{158455} a^{14} - \frac{3989301}{158455} a^{13} - \frac{4422977}{158455} a^{12} + \frac{3378487}{158455} a^{11} + \frac{4404977}{158455} a^{10} - \frac{101333}{3685} a^{9} - \frac{1842426}{158455} a^{8} + \frac{5974183}{158455} a^{7} + \frac{1953703}{158455} a^{6} - \frac{809525}{31691} a^{5} - \frac{2610266}{158455} a^{4} + \frac{256174}{31691} a^{3} + \frac{172506}{31691} a^{2} - \frac{72504}{158455} a - \frac{2793}{31691} \),  \( \frac{18105}{31691} a^{19} - \frac{106041}{158455} a^{18} - \frac{478697}{158455} a^{17} + \frac{82952}{158455} a^{16} + \frac{434098}{31691} a^{15} - \frac{43147}{158455} a^{14} - \frac{4057543}{158455} a^{13} - \frac{206512}{31691} a^{12} + \frac{4695331}{158455} a^{11} + \frac{1383621}{158455} a^{10} - \frac{4807871}{158455} a^{9} + \frac{1202912}{158455} a^{8} + \frac{869081}{31691} a^{7} - \frac{1436499}{158455} a^{6} - \frac{3815667}{158455} a^{5} - \frac{413966}{158455} a^{4} + \frac{4358}{335} a^{3} + \frac{360374}{158455} a^{2} - \frac{182552}{158455} a - \frac{39815}{31691} \),  \( \frac{5481}{14405} a^{19} - \frac{17642}{31691} a^{18} - \frac{58019}{31691} a^{17} + \frac{1633}{2365} a^{16} + \frac{6613}{737} a^{15} - \frac{340836}{158455} a^{14} - \frac{2492584}{158455} a^{13} - \frac{395193}{158455} a^{12} + \frac{281881}{14405} a^{11} + \frac{371717}{158455} a^{10} - \frac{585570}{31691} a^{9} + \frac{235819}{31691} a^{8} + \frac{217612}{14405} a^{7} - \frac{24952}{3685} a^{6} - \frac{2272144}{158455} a^{5} + \frac{95556}{158455} a^{4} + \frac{224948}{31691} a^{3} + \frac{137956}{158455} a^{2} - \frac{127768}{158455} a - \frac{179103}{158455} \),  \( \frac{1383}{31691} a^{19} + \frac{43027}{158455} a^{18} + \frac{4461}{31691} a^{17} - \frac{249489}{158455} a^{16} - \frac{35872}{14405} a^{15} + \frac{562138}{158455} a^{14} + \frac{327816}{31691} a^{13} + \frac{9142}{31691} a^{12} - \frac{2416262}{158455} a^{11} - \frac{1144758}{158455} a^{10} + \frac{1566863}{158455} a^{9} + \frac{68545}{31691} a^{8} - \frac{301245}{31691} a^{7} + \frac{798328}{158455} a^{6} + \frac{2682113}{158455} a^{5} + \frac{41286}{14405} a^{4} - \frac{282814}{31691} a^{3} - \frac{928293}{158455} a^{2} + \frac{54361}{31691} a + \frac{42072}{31691} \),  \( \frac{87209}{158455} a^{19} - \frac{314}{3685} a^{18} - \frac{420493}{158455} a^{17} - \frac{438201}{158455} a^{16} + \frac{1470533}{158455} a^{15} + \frac{1594074}{158455} a^{14} - \frac{1374084}{158455} a^{13} - \frac{3242878}{158455} a^{12} + \frac{267086}{158455} a^{11} + \frac{2680544}{158455} a^{10} - \frac{576377}{158455} a^{9} - \frac{1257601}{158455} a^{8} + \frac{1566077}{158455} a^{7} + \frac{2747948}{158455} a^{6} - \frac{789176}{158455} a^{5} - \frac{535495}{31691} a^{4} - \frac{100326}{31691} a^{3} + \frac{122938}{31691} a^{2} + \frac{99959}{31691} a + \frac{597}{2365} \),  \( \frac{4484}{31691} a^{19} - \frac{101161}{158455} a^{18} - \frac{46089}{158455} a^{17} + \frac{9242}{3685} a^{16} + \frac{564269}{158455} a^{15} - \frac{1643278}{158455} a^{14} - \frac{238442}{31691} a^{13} + \frac{222176}{14405} a^{12} + \frac{2187186}{158455} a^{11} - \frac{2633957}{158455} a^{10} - \frac{1965017}{158455} a^{9} + \frac{325248}{14405} a^{8} - \frac{39564}{158455} a^{7} - \frac{3440854}{158455} a^{6} - \frac{25436}{31691} a^{5} + \frac{34461}{2365} a^{4} + \frac{437889}{158455} a^{3} - \frac{857719}{158455} a^{2} + \frac{141286}{158455} a - \frac{1001}{14405} \),  \( \frac{182152}{158455} a^{19} - \frac{238}{737} a^{18} - \frac{834859}{158455} a^{17} - \frac{138950}{31691} a^{16} + \frac{2961408}{158455} a^{15} + \frac{200086}{14405} a^{14} - \frac{3047967}{158455} a^{13} - \frac{3760729}{158455} a^{12} + \frac{2156942}{158455} a^{11} + \frac{1974783}{158455} a^{10} - \frac{4076662}{158455} a^{9} + \frac{300027}{158455} a^{8} + \frac{4253006}{158455} a^{7} + \frac{2095898}{158455} a^{6} - \frac{1291494}{158455} a^{5} - \frac{1275702}{158455} a^{4} + \frac{16472}{31691} a^{3} - \frac{25139}{14405} a^{2} + \frac{52000}{31691} a + \frac{59932}{158455} \),  \( \frac{13892}{158455} a^{19} - \frac{2038}{3685} a^{18} - \frac{32196}{158455} a^{17} + \frac{334272}{158455} a^{16} + \frac{500196}{158455} a^{15} - \frac{111526}{14405} a^{14} - \frac{1120299}{158455} a^{13} + \frac{1246232}{158455} a^{12} + \frac{1669156}{158455} a^{11} - \frac{836989}{158455} a^{10} - \frac{1051869}{158455} a^{9} + \frac{1881993}{158455} a^{8} - \frac{268198}{158455} a^{7} - \frac{1608593}{158455} a^{6} - \frac{780173}{158455} a^{5} + \frac{311659}{158455} a^{4} + \frac{155630}{31691} a^{3} + \frac{3409}{14405} a^{2} + \frac{210953}{158455} a - \frac{230939}{158455} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 630.365304677 \)
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

\(\Q(\sqrt{-47}) \), \(\Q(\sqrt{141}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-47})\), 5.1.2209.1 x5, 10.0.229345007.1, 10.2.55730836701.1 x5, 10.0.1185762483.1 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$47$47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$