Properties

Label 20.0.60220746496...1001.1
Degree $20$
Signature $[0, 10]$
Discriminant $47^{10}\cdot 107^{2}$
Root discriminant $10.94$
Ramified primes $47, 107$
Class number $1$
Class group Trivial
Galois Group $C_2\times C_2^4:D_5$ (as 20T81)

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Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

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

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:  \(602207464968018231001=47^{10}\cdot 107^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.94$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $47, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{29} a^{18} + \frac{10}{29} a^{17} - \frac{12}{29} a^{16} + \frac{3}{29} a^{15} + \frac{2}{29} a^{14} - \frac{14}{29} a^{13} - \frac{1}{29} a^{12} + \frac{14}{29} a^{11} - \frac{9}{29} a^{10} + \frac{4}{29} a^{9} - \frac{9}{29} a^{8} + \frac{14}{29} a^{7} - \frac{1}{29} a^{6} - \frac{14}{29} a^{5} + \frac{2}{29} a^{4} + \frac{3}{29} a^{3} - \frac{12}{29} a^{2} + \frac{10}{29} a + \frac{1}{29}$, $\frac{1}{377} a^{19} - \frac{4}{377} a^{18} + \frac{138}{377} a^{17} + \frac{2}{29} a^{16} - \frac{69}{377} a^{15} + \frac{74}{377} a^{14} - \frac{14}{29} a^{13} - \frac{146}{377} a^{12} + \frac{27}{377} a^{11} - \frac{102}{377} a^{10} + \frac{80}{377} a^{9} + \frac{53}{377} a^{8} - \frac{81}{377} a^{7} - \frac{121}{377} a^{5} + \frac{178}{377} a^{4} + \frac{7}{29} a^{3} - \frac{83}{377} a^{2} + \frac{35}{377} a + \frac{131}{377}$

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:  \( \frac{245}{377} a^{19} - \frac{343}{377} a^{18} - \frac{159}{377} a^{17} - \frac{11}{29} a^{16} + \frac{1217}{377} a^{15} + \frac{554}{377} a^{14} + \frac{31}{29} a^{13} - \frac{2854}{377} a^{12} - \frac{2186}{377} a^{11} - \frac{940}{377} a^{10} + \frac{3675}{377} a^{9} + \frac{2728}{377} a^{8} + \frac{1514}{377} a^{7} - \frac{194}{29} a^{6} - \frac{1994}{377} a^{5} - \frac{1487}{377} a^{4} + \frac{93}{29} a^{3} + \frac{296}{377} a^{2} + \frac{619}{377} a - \frac{444}{377} \),  \( \frac{96}{377} a^{19} - \frac{202}{377} a^{18} + \frac{365}{377} a^{17} - \frac{63}{29} a^{16} + \frac{1085}{377} a^{15} - \frac{449}{377} a^{14} + \frac{113}{29} a^{13} - \frac{1757}{377} a^{12} + \frac{239}{377} a^{11} - \frac{2382}{377} a^{10} + \frac{1999}{377} a^{9} + \frac{434}{377} a^{8} + \frac{2689}{377} a^{7} - \frac{159}{29} a^{6} - \frac{215}{377} a^{5} - \frac{2152}{377} a^{4} + \frac{76}{29} a^{3} + \frac{27}{377} a^{2} + \frac{1033}{377} a - \frac{437}{377} \),  \( a \),  \( \frac{269}{377} a^{19} - \frac{582}{377} a^{18} + \frac{215}{377} a^{17} - \frac{5}{29} a^{16} + \frac{1017}{377} a^{15} + \frac{159}{377} a^{14} - \frac{6}{29} a^{13} - \frac{2445}{377} a^{12} - \frac{901}{377} a^{11} + \frac{538}{377} a^{10} + \frac{3138}{377} a^{9} + \frac{1517}{377} a^{8} + \frac{207}{377} a^{7} - \frac{183}{29} a^{6} - \frac{1765}{377} a^{5} - \frac{140}{377} a^{4} + \frac{83}{29} a^{3} + \frac{774}{377} a^{2} + \frac{14}{13} a - \frac{82}{377} \),  \( \frac{254}{377} a^{19} - \frac{600}{377} a^{18} + \frac{758}{377} a^{17} - \frac{50}{29} a^{16} + \frac{1064}{377} a^{15} - \frac{730}{377} a^{14} + \frac{143}{29} a^{13} - \frac{1308}{377} a^{12} + \frac{1749}{377} a^{11} - \frac{1754}{377} a^{10} + \frac{118}{377} a^{9} - \frac{2346}{377} a^{8} + \frac{2215}{377} a^{7} - \frac{3}{29} a^{6} + \frac{2273}{377} a^{5} - \frac{704}{377} a^{4} + \frac{18}{29} a^{3} - \frac{1192}{377} a^{2} + \frac{21}{13} a - \frac{240}{377} \),  \( \frac{210}{377} a^{19} - \frac{190}{377} a^{18} - \frac{335}{377} a^{17} + \frac{23}{29} a^{16} + \frac{278}{377} a^{15} + \frac{1383}{377} a^{14} - \frac{15}{29} a^{13} - \frac{773}{377} a^{12} - \frac{2572}{377} a^{11} - \frac{126}{377} a^{10} + \frac{550}{377} a^{9} + \frac{2264}{377} a^{8} + \frac{384}{377} a^{7} + \frac{37}{29} a^{6} - \frac{46}{13} a^{5} + \frac{226}{377} a^{4} - \frac{62}{29} a^{3} + \frac{1}{13} a^{2} - \frac{99}{377} a + \frac{262}{377} \),  \( \frac{266}{377} a^{19} - \frac{63}{377} a^{18} - \frac{407}{377} a^{17} + \frac{14}{29} a^{16} + \frac{106}{377} a^{15} + \frac{2082}{377} a^{14} + \frac{99}{29} a^{13} + \frac{502}{377} a^{12} - \frac{2932}{377} a^{11} - \frac{2211}{377} a^{10} - \frac{1860}{377} a^{9} + \frac{2073}{377} a^{8} + \frac{2647}{377} a^{7} + \frac{213}{29} a^{6} - \frac{583}{377} a^{5} - \frac{414}{377} a^{4} - \frac{111}{29} a^{3} - \frac{160}{377} a^{2} + \frac{93}{377} a + \frac{409}{377} \),  \( \frac{100}{377} a^{19} - \frac{309}{377} a^{18} + \frac{384}{377} a^{17} - a^{16} + \frac{536}{377} a^{15} - \frac{335}{377} a^{14} + \frac{39}{29} a^{13} - \frac{1119}{377} a^{12} + \frac{204}{377} a^{11} - \frac{840}{377} a^{10} + \frac{1201}{377} a^{9} + \frac{334}{377} a^{8} + \frac{1468}{377} a^{7} - \frac{94}{29} a^{6} - \frac{556}{377} a^{5} - \frac{1245}{377} a^{4} + \frac{25}{29} a^{3} + \frac{410}{377} a^{2} + \frac{640}{377} a - \frac{4}{377} \),  \( \frac{45}{377} a^{19} - \frac{193}{377} a^{18} + \frac{48}{377} a^{17} + \frac{15}{29} a^{16} + \frac{249}{377} a^{15} - \frac{466}{377} a^{14} - \frac{36}{29} a^{13} - \frac{902}{377} a^{12} + \frac{279}{377} a^{11} + \frac{1559}{377} a^{10} + \frac{1286}{377} a^{9} - \frac{137}{377} a^{8} - \frac{1188}{377} a^{7} - \frac{86}{29} a^{6} - \frac{362}{377} a^{5} + \frac{444}{377} a^{4} + \frac{51}{29} a^{3} - \frac{186}{377} a^{2} - \frac{63}{377} a + \frac{227}{377} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 79.7167288943 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2\times C_2^4:D_5$ (as 20T81):

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

\(\Q(\sqrt{-47}) \), 5.1.2209.1 x5, 10.2.24539915749.1, 10.0.522125867.1, 10.0.229345007.1

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{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}$
$107$107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
107.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$