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Properties

Label 20.0.10946054092...0625.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 11^{18}\cdot 17^{10}$
Root discriminant $79.79$
Ramified primes $5, 11, 17$
Class number $17416$ (GRH)
Class group $[17416]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![154880969, -30430726, -21459284, -46370542, 26804764, 10868044, -2328113, -3332126, 349107, 682712, 8986, -102854, -4180, 11878, 1224, -1320, -26, 72, 11, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 11*x^18 + 72*x^17 - 26*x^16 - 1320*x^15 + 1224*x^14 + 11878*x^13 - 4180*x^12 - 102854*x^11 + 8986*x^10 + 682712*x^9 + 349107*x^8 - 3332126*x^7 - 2328113*x^6 + 10868044*x^5 + 26804764*x^4 - 46370542*x^3 - 21459284*x^2 - 30430726*x + 154880969)
 
gp: K = bnfinit(x^20 - 8*x^19 + 11*x^18 + 72*x^17 - 26*x^16 - 1320*x^15 + 1224*x^14 + 11878*x^13 - 4180*x^12 - 102854*x^11 + 8986*x^10 + 682712*x^9 + 349107*x^8 - 3332126*x^7 - 2328113*x^6 + 10868044*x^5 + 26804764*x^4 - 46370542*x^3 - 21459284*x^2 - 30430726*x + 154880969, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 11 x^{18} + 72 x^{17} - 26 x^{16} - 1320 x^{15} + 1224 x^{14} + 11878 x^{13} - 4180 x^{12} - 102854 x^{11} + 8986 x^{10} + 682712 x^{9} + 349107 x^{8} - 3332126 x^{7} - 2328113 x^{6} + 10868044 x^{5} + 26804764 x^{4} - 46370542 x^{3} - 21459284 x^{2} - 30430726 x + 154880969 \)

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:  \(109460540927745402703534939794619140625=5^{10}\cdot 11^{18}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.79$
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:  $5, 11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(935=5\cdot 11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{935}(256,·)$, $\chi_{935}(1,·)$, $\chi_{935}(579,·)$, $\chi_{935}(526,·)$, $\chi_{935}(16,·)$, $\chi_{935}(849,·)$, $\chi_{935}(851,·)$, $\chi_{935}(84,·)$, $\chi_{935}(86,·)$, $\chi_{935}(919,·)$, $\chi_{935}(409,·)$, $\chi_{935}(356,·)$, $\chi_{935}(934,·)$, $\chi_{935}(679,·)$, $\chi_{935}(424,·)$, $\chi_{935}(494,·)$, $\chi_{935}(239,·)$, $\chi_{935}(696,·)$, $\chi_{935}(441,·)$, $\chi_{935}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{178} a^{12} + \frac{15}{178} a^{11} + \frac{10}{89} a^{10} + \frac{37}{89} a^{9} + \frac{4}{89} a^{8} + \frac{43}{89} a^{7} + \frac{11}{178} a^{6} + \frac{85}{178} a^{5} + \frac{4}{89} a^{4} - \frac{32}{89} a^{3} + \frac{49}{178} a^{2} - \frac{44}{89} a + \frac{31}{89}$, $\frac{1}{178} a^{13} - \frac{27}{178} a^{11} + \frac{41}{178} a^{10} + \frac{55}{178} a^{9} - \frac{17}{89} a^{8} - \frac{33}{178} a^{7} - \frac{40}{89} a^{6} - \frac{21}{178} a^{5} + \frac{83}{178} a^{4} + \frac{15}{89} a^{3} + \frac{67}{178} a^{2} - \frac{21}{89} a + \frac{49}{178}$, $\frac{1}{178} a^{14} + \frac{1}{178} a^{11} - \frac{14}{89} a^{10} + \frac{3}{89} a^{9} + \frac{5}{178} a^{8} - \frac{36}{89} a^{7} - \frac{40}{89} a^{6} - \frac{25}{178} a^{5} - \frac{21}{178} a^{4} - \frac{59}{178} a^{3} + \frac{35}{178} a^{2} + \frac{38}{89} a + \frac{36}{89}$, $\frac{1}{178} a^{15} - \frac{43}{178} a^{11} - \frac{7}{89} a^{10} - \frac{69}{178} a^{9} - \frac{40}{89} a^{8} + \frac{6}{89} a^{7} - \frac{18}{89} a^{6} + \frac{36}{89} a^{5} - \frac{67}{178} a^{4} - \frac{79}{178} a^{3} + \frac{27}{178} a^{2} - \frac{9}{89} a - \frac{31}{89}$, $\frac{1}{178} a^{16} + \frac{4}{89} a^{11} - \frac{5}{89} a^{10} + \frac{38}{89} a^{9} - \frac{38}{89} a^{7} + \frac{11}{178} a^{6} - \frac{61}{178} a^{5} - \frac{1}{89} a^{4} - \frac{55}{178} a^{3} - \frac{47}{178} a^{2} - \frac{19}{178} a - \frac{2}{89}$, $\frac{1}{178} a^{17} - \frac{41}{178} a^{11} + \frac{5}{178} a^{10} - \frac{29}{89} a^{9} + \frac{19}{89} a^{8} + \frac{35}{178} a^{7} + \frac{29}{178} a^{6} - \frac{59}{178} a^{5} - \frac{15}{89} a^{4} - \frac{69}{178} a^{3} - \frac{55}{178} a^{2} + \frac{77}{178} a + \frac{19}{89}$, $\frac{1}{7654} a^{18} + \frac{21}{7654} a^{17} - \frac{8}{3827} a^{16} + \frac{6}{3827} a^{15} + \frac{3}{3827} a^{14} - \frac{3}{3827} a^{13} - \frac{7}{3827} a^{12} + \frac{1743}{7654} a^{11} - \frac{852}{3827} a^{10} + \frac{2307}{7654} a^{9} + \frac{1213}{7654} a^{8} - \frac{1721}{3827} a^{7} - \frac{1363}{7654} a^{6} - \frac{1160}{3827} a^{5} + \frac{84}{3827} a^{4} + \frac{2663}{7654} a^{3} - \frac{2075}{7654} a^{2} + \frac{3279}{7654} a - \frac{11}{178}$, $\frac{1}{4813483962189316240363870000172764486052518028147325937238} a^{19} + \frac{120845376510118581276718998034028692976678508485397771}{2406741981094658120181935000086382243026259014073662968619} a^{18} - \frac{6686170562651222132370628324074963422228653832396294213}{4813483962189316240363870000172764486052518028147325937238} a^{17} - \frac{6676516120220507759644507818416901290528905339162699727}{2406741981094658120181935000086382243026259014073662968619} a^{16} - \frac{5967649045898218639227282292497248050253668516159719621}{4813483962189316240363870000172764486052518028147325937238} a^{15} + \frac{9300952843713327788633192151769511650072610182769318785}{4813483962189316240363870000172764486052518028147325937238} a^{14} + \frac{638370824555146878778238300933979025882736976350565806}{2406741981094658120181935000086382243026259014073662968619} a^{13} - \frac{9738903635146989077719473622624497875621040228089353551}{4813483962189316240363870000172764486052518028147325937238} a^{12} + \frac{1116987488246679898217748221580881017744424216205493160563}{4813483962189316240363870000172764486052518028147325937238} a^{11} - \frac{188317367427655415645473171438028607332837375108719534616}{2406741981094658120181935000086382243026259014073662968619} a^{10} - \frac{325508917987561471755225930759450830628327491425090276050}{2406741981094658120181935000086382243026259014073662968619} a^{9} - \frac{219553483734022487894353922692310190547209363966191568471}{2406741981094658120181935000086382243026259014073662968619} a^{8} - \frac{910454846273941545368270448912384393905135305994315635509}{4813483962189316240363870000172764486052518028147325937238} a^{7} + \frac{1727285311359140950255316335343565531191652600802243561679}{4813483962189316240363870000172764486052518028147325937238} a^{6} + \frac{25113280698237322368626211276373441091234280231159833239}{111941487492774796287531860469134057815174837863891300866} a^{5} - \frac{1062765309976449684001266881301029429546780556894750178414}{2406741981094658120181935000086382243026259014073662968619} a^{4} + \frac{2072232471775642481797601375077004741408835120352690233065}{4813483962189316240363870000172764486052518028147325937238} a^{3} + \frac{86689399631720732750562334943273801853259204873618535333}{2406741981094658120181935000086382243026259014073662968619} a^{2} - \frac{392117978129323185955077400060287692899898177428562311730}{2406741981094658120181935000086382243026259014073662968619} a + \frac{7748163587697056036379684512244295252627538162790234835}{55970743746387398143765930234567028907587418931945650433}$

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

Class group and class number

$C_{17416}$, which has order $17416$ (assuming GRH)

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:  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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3338983.62101 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-935}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{17}, \sqrt{-55})\), \(\Q(\zeta_{11})^+\), 10.0.10462339170938084375.1, 10.0.7368586534375.1, 10.10.304358957700017.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
5Data not computed
11Data not computed
$17$17.10.5.1$x^{10} - 578 x^{6} + 83521 x^{2} - 51114852$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
17.10.5.1$x^{10} - 578 x^{6} + 83521 x^{2} - 51114852$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$