Properties

Label 20.0.29411962588...3449.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{18}\cdot 23^{2}$
Root discriminant $11.84$
Ramified primes $11, 23$
Class number $1$
Class group Trivial
Galois Group $C_2^2\times C_2^4:C_5$ (as 20T86)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 24, -72, 177, -363, 637, -973, 1307, -1556, 1649, -1556, 1307, -973, 637, -363, 177, -72, 24, -6, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 24*x^18 - 72*x^17 + 177*x^16 - 363*x^15 + 637*x^14 - 973*x^13 + 1307*x^12 - 1556*x^11 + 1649*x^10 - 1556*x^9 + 1307*x^8 - 973*x^7 + 637*x^6 - 363*x^5 + 177*x^4 - 72*x^3 + 24*x^2 - 6*x + 1)
gp: K = bnfinit(x^20 - 6*x^19 + 24*x^18 - 72*x^17 + 177*x^16 - 363*x^15 + 637*x^14 - 973*x^13 + 1307*x^12 - 1556*x^11 + 1649*x^10 - 1556*x^9 + 1307*x^8 - 973*x^7 + 637*x^6 - 363*x^5 + 177*x^4 - 72*x^3 + 24*x^2 - 6*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 6 x^{19} \) \(\mathstrut +\mathstrut 24 x^{18} \) \(\mathstrut -\mathstrut 72 x^{17} \) \(\mathstrut +\mathstrut 177 x^{16} \) \(\mathstrut -\mathstrut 363 x^{15} \) \(\mathstrut +\mathstrut 637 x^{14} \) \(\mathstrut -\mathstrut 973 x^{13} \) \(\mathstrut +\mathstrut 1307 x^{12} \) \(\mathstrut -\mathstrut 1556 x^{11} \) \(\mathstrut +\mathstrut 1649 x^{10} \) \(\mathstrut -\mathstrut 1556 x^{9} \) \(\mathstrut +\mathstrut 1307 x^{8} \) \(\mathstrut -\mathstrut 973 x^{7} \) \(\mathstrut +\mathstrut 637 x^{6} \) \(\mathstrut -\mathstrut 363 x^{5} \) \(\mathstrut +\mathstrut 177 x^{4} \) \(\mathstrut -\mathstrut 72 x^{3} \) \(\mathstrut +\mathstrut 24 x^{2} \) \(\mathstrut -\mathstrut 6 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(2941196258837390453449=11^{18}\cdot 23^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.84$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{109} a^{19} - \frac{7}{109} a^{18} + \frac{31}{109} a^{17} + \frac{6}{109} a^{16} - \frac{47}{109} a^{15} + \frac{11}{109} a^{14} - \frac{28}{109} a^{13} + \frac{36}{109} a^{12} - \frac{37}{109} a^{11} + \frac{7}{109} a^{10} + \frac{7}{109} a^{9} - \frac{37}{109} a^{8} + \frac{36}{109} a^{7} - \frac{28}{109} a^{6} + \frac{11}{109} a^{5} - \frac{47}{109} a^{4} + \frac{6}{109} a^{3} + \frac{31}{109} a^{2} - \frac{7}{109} a + \frac{1}{109}$

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:  \( \frac{20}{109} a^{19} - \frac{140}{109} a^{18} + \frac{511}{109} a^{17} - \frac{1515}{109} a^{16} + \frac{3638}{109} a^{15} - \frac{7410}{109} a^{14} + \frac{12956}{109} a^{13} - \frac{20208}{109} a^{12} + \frac{28254}{109} a^{11} - \frac{34849}{109} a^{10} + \frac{38399}{109} a^{9} - \frac{37364}{109} a^{8} + \frac{31894}{109} a^{7} - \frac{23995}{109} a^{6} + \frac{15480}{109} a^{5} - \frac{8461}{109} a^{4} + \frac{4153}{109} a^{3} - \frac{1669}{109} a^{2} + \frac{623}{109} a - \frac{198}{109} \) (order $22$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{285}{109} a^{19} - \frac{1341}{109} a^{18} + \frac{4693}{109} a^{17} - \frac{12024}{109} a^{16} + \frac{25191}{109} a^{15} - \frac{41882}{109} a^{14} + \frac{56548}{109} a^{13} - \frac{60481}{109} a^{12} + \frac{46135}{109} a^{11} - \frac{14028}{109} a^{10} - \frac{26999}{109} a^{9} + \frac{64120}{109} a^{8} - \frac{84025}{109} a^{7} + \frac{83580}{109} a^{6} - \frac{67061}{109} a^{5} + \frac{43830}{109} a^{4} - \frac{23469}{109} a^{3} + \frac{10034}{109} a^{2} - \frac{3085}{109} a + \frac{721}{109} \),  \( \frac{105}{109} a^{19} - \frac{626}{109} a^{18} + \frac{2492}{109} a^{17} - \frac{7654}{109} a^{16} + \frac{19263}{109} a^{15} - \frac{40919}{109} a^{14} + \frac{74777}{109} a^{13} - \frac{120044}{109} a^{12} + \frac{169425}{109} a^{11} - \frac{211488}{109} a^{10} + \frac{233668}{109} a^{9} - \frac{227771}{109} a^{8} + \frac{195293}{109} a^{7} - \frac{146275}{109} a^{6} + \frac{94132}{109} a^{5} - \frac{51369}{109} a^{4} + \frac{22975}{109} a^{3} - \frac{8081}{109} a^{2} + \frac{2317}{109} a - \frac{440}{109} \),  \( \frac{114}{109} a^{19} - \frac{798}{109} a^{18} + \frac{3207}{109} a^{17} - \frac{9671}{109} a^{16} + \frac{23309}{109} a^{15} - \frac{46597}{109} a^{14} + \frac{77904}{109} a^{13} - \frac{111545}{109} a^{12} + \frac{137918}{109} a^{11} - \frac{148532}{109} a^{10} + \frac{139337}{109} a^{9} - \frac{113436}{109} a^{8} + \frac{79314}{109} a^{7} - \frac{47010}{109} a^{6} + \frac{22727}{109} a^{5} - \frac{8410}{109} a^{4} + \frac{1992}{109} a^{3} - \frac{63}{109} a^{2} - \frac{144}{109} a + \frac{5}{109} \),  \( \frac{15}{109} a^{19} + \frac{4}{109} a^{18} - \frac{407}{109} a^{17} + \frac{2161}{109} a^{16} - \frac{7572}{109} a^{15} + \frac{20112}{109} a^{14} - \frac{43475}{109} a^{13} + \frac{77385}{109} a^{12} - \frac{116967}{109} a^{11} + \frac{152051}{109} a^{10} - \frac{171679}{109} a^{9} + \frac{169158}{109} a^{8} - \frac{145411}{109} a^{7} + \frac{108471}{109} a^{6} - \frac{69813}{109} a^{5} + \frac{38099}{109} a^{4} - \frac{17241}{109} a^{3} + \frac{6133}{109} a^{2} - \frac{1631}{109} a + \frac{233}{109} \),  \( \frac{23}{109} a^{19} - \frac{379}{109} a^{18} + \frac{2021}{109} a^{17} - \frac{7274}{109} a^{16} + \frac{19956}{109} a^{15} - \frac{44655}{109} a^{14} + \frac{82414}{109} a^{13} - \frac{127465}{109} a^{12} + \frac{167118}{109} a^{11} - \frac{187864}{109} a^{10} + \frac{181101}{109} a^{9} - \frac{149636}{109} a^{8} + \frac{104705}{109} a^{7} - \frac{60812}{109} a^{6} + \frac{28484}{109} a^{5} - \frac{10346}{109} a^{4} + \frac{2645}{109} a^{3} - \frac{595}{109} a^{2} + \frac{57}{109} a + \frac{23}{109} \),  \( \frac{20}{109} a^{19} - \frac{249}{109} a^{18} + \frac{1056}{109} a^{17} - \frac{3477}{109} a^{16} + \frac{8870}{109} a^{15} - \frac{18964}{109} a^{14} + \frac{33775}{109} a^{13} - \frac{52036}{109} a^{12} + \frac{70110}{109} a^{11} - \frac{82809}{109} a^{10} + \frac{86359}{109} a^{9} - \frac{79329}{109} a^{8} + \frac{63613}{109} a^{7} - \frac{44814}{109} a^{6} + \frac{27034}{109} a^{5} - \frac{13802}{109} a^{4} + \frac{6006}{109} a^{3} - \frac{2214}{109} a^{2} + \frac{732}{109} a - \frac{198}{109} \),  \( \frac{288}{109} a^{19} - \frac{1253}{109} a^{18} + \frac{4241}{109} a^{17} - \frac{10589}{109} a^{16} + \frac{22216}{109} a^{15} - \frac{37816}{109} a^{14} + \frac{55592}{109} a^{13} - \frac{72581}{109} a^{12} + \frac{85373}{109} a^{11} - \frac{92051}{109} a^{10} + \frac{90960}{109} a^{9} - \frac{81615}{109} a^{8} + \frac{66721}{109} a^{7} - \frac{48285}{109} a^{6} + \frac{30200}{109} a^{5} - \frac{15825}{109} a^{4} + \frac{6633}{109} a^{3} - \frac{2081}{109} a^{2} + \frac{600}{109} a - \frac{39}{109} \),  \( \frac{134}{109} a^{19} - \frac{720}{109} a^{18} + \frac{2955}{109} a^{17} - \frac{8897}{109} a^{16} + \frac{22260}{109} a^{15} - \frac{46159}{109} a^{14} + \frac{82358}{109} a^{13} - \frac{126739}{109} a^{12} + \frac{169769}{109} a^{11} - \frac{199077}{109} a^{10} + \frac{204986}{109} a^{9} - \frac{184590}{109} a^{8} + \frac{145325}{109} a^{7} - \frac{98473}{109} a^{6} + \frac{56737}{109} a^{5} - \frac{27008}{109} a^{4} + \frac{10069}{109} a^{3} - \frac{2604}{109} a^{2} + \frac{370}{109} a + \frac{134}{109} \),  \( \frac{28}{109} a^{19} + \frac{131}{109} a^{18} - \frac{767}{109} a^{17} + \frac{3111}{109} a^{16} - \frac{8619}{109} a^{15} + \frac{19710}{109} a^{14} - \frac{36100}{109} a^{13} + \frac{56053}{109} a^{12} - \frac{75810}{109} a^{11} + \frac{90339}{109} a^{10} - \frac{96814}{109} a^{9} + \frac{93576}{109} a^{8} - \frac{81505}{109} a^{7} + \frac{64398}{109} a^{6} - \frac{45472}{109} a^{5} + \frac{28223}{109} a^{4} - \frac{14983}{109} a^{3} + \frac{6427}{109} a^{2} - \frac{2049}{109} a + \frac{573}{109} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 2189.13991382 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T86):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{11})\), 10.6.54232796893.1, 10.4.4930254263.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
11Data not computed
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$