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Properties

Label 20.0.22568047146...4237.1
Degree $20$
Signature $[0, 10]$
Discriminant $7^{15}\cdot 11^{9}\cdot 17^{10}$
Root discriminant $52.20$
Ramified primes $7, 11, 17$
Class number Not computed
Class group Not computed
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![395131293061, 185883345808, 52659896224, -10400245765, 789030112, -719171544, -64052500, 88733076, 34025996, -9062519, -557925, -356967, 124321, 6363, 11855, -2748, 199, -54, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 54*x^17 + 199*x^16 - 2748*x^15 + 11855*x^14 + 6363*x^13 + 124321*x^12 - 356967*x^11 - 557925*x^10 - 9062519*x^9 + 34025996*x^8 + 88733076*x^7 - 64052500*x^6 - 719171544*x^5 + 789030112*x^4 - 10400245765*x^3 + 52659896224*x^2 + 185883345808*x + 395131293061)
 
gp: K = bnfinit(x^20 - 2*x^19 - 54*x^17 + 199*x^16 - 2748*x^15 + 11855*x^14 + 6363*x^13 + 124321*x^12 - 356967*x^11 - 557925*x^10 - 9062519*x^9 + 34025996*x^8 + 88733076*x^7 - 64052500*x^6 - 719171544*x^5 + 789030112*x^4 - 10400245765*x^3 + 52659896224*x^2 + 185883345808*x + 395131293061, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 54 x^{17} + 199 x^{16} - 2748 x^{15} + 11855 x^{14} + 6363 x^{13} + 124321 x^{12} - 356967 x^{11} - 557925 x^{10} - 9062519 x^{9} + 34025996 x^{8} + 88733076 x^{7} - 64052500 x^{6} - 719171544 x^{5} + 789030112 x^{4} - 10400245765 x^{3} + 52659896224 x^{2} + 185883345808 x + 395131293061 \)

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:  \(22568047146271109644424925266534237=7^{15}\cdot 11^{9}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.20$
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:  $7, 11, 17$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{16} a^{17} - \frac{1}{16} a^{16} + \frac{1}{4} a^{15} - \frac{1}{4} a^{14} + \frac{1}{4} a^{13} + \frac{7}{16} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{3}{8} a^{8} + \frac{5}{16} a^{7} + \frac{1}{8} a^{6} - \frac{7}{16} a^{5} + \frac{3}{8} a^{4} + \frac{5}{16} a^{3} + \frac{3}{8} a^{2} + \frac{1}{16} a - \frac{7}{16}$, $\frac{1}{64} a^{18} + \frac{3}{64} a^{16} + \frac{1}{4} a^{15} - \frac{1}{4} a^{14} + \frac{5}{16} a^{13} + \frac{7}{64} a^{12} - \frac{27}{64} a^{11} + \frac{1}{8} a^{9} - \frac{21}{64} a^{8} - \frac{9}{64} a^{7} - \frac{5}{64} a^{6} - \frac{17}{64} a^{5} - \frac{5}{64} a^{4} - \frac{21}{64} a^{3} - \frac{25}{64} a^{2} + \frac{5}{32} a - \frac{23}{64}$, $\frac{1}{81636919227060523784733160783990832603520886938889498622083609906122386962903836434825462819519861176891136} a^{19} - \frac{119317299249676775195368070127154216452153733691840129643984878094365411183387053565540088026019947333643}{81636919227060523784733160783990832603520886938889498622083609906122386962903836434825462819519861176891136} a^{18} - \frac{23358492299817059690257518681160289803252195873400969626584478068266337546207990612834240157779661884445}{81636919227060523784733160783990832603520886938889498622083609906122386962903836434825462819519861176891136} a^{17} + \frac{5416664515388633889805642154141336014074264792546841717852913439697648665993395871231212553827943063308815}{81636919227060523784733160783990832603520886938889498622083609906122386962903836434825462819519861176891136} a^{16} + \frac{95526994063842491313272520203967177865744384545094161238325404584578721186797191891280866162734654941567}{637788431461410342068227818624928379715006929210074207985028202391581148147686222147073928277498915444462} a^{15} - \frac{7332501259959014871911271203592902976591418793940127237348362909057329393912809827009626828020271270486479}{20409229806765130946183290195997708150880221734722374655520902476530596740725959108706365704879965294222784} a^{14} + \frac{25717069120024508079485773192452986713464741930959974743825789083833723455832239769441915810112592530663787}{81636919227060523784733160783990832603520886938889498622083609906122386962903836434825462819519861176891136} a^{13} + \frac{4539112424688237595563344547036694507756847008530921152704990414947939491633982912221481262150778956551107}{10204614903382565473091645097998854075440110867361187327760451238265298370362979554353182852439982647111392} a^{12} - \frac{38909529361896942419352641366084364889976410551131727575420924804435110199221884543042959321666798628078519}{81636919227060523784733160783990832603520886938889498622083609906122386962903836434825462819519861176891136} a^{11} + \frac{1583328466909744285527123606628617270538796672534954456337309678656570270953951772455248538751431444790633}{10204614903382565473091645097998854075440110867361187327760451238265298370362979554353182852439982647111392} a^{10} + \frac{40095712211060731735813829432831805764925322337809134122458446671923004102537180235416433204683943693076627}{81636919227060523784733160783990832603520886938889498622083609906122386962903836434825462819519861176891136} a^{9} - \frac{18130848617649140931734083173008227873479432003264300787239148672786238865821679050884156495492615187631889}{40818459613530261892366580391995416301760443469444749311041804953061193481451918217412731409759930588445568} a^{8} - \frac{10932868921099300619144163487189186038430568496165036027301440466266556242176467789532164578544881936722657}{40818459613530261892366580391995416301760443469444749311041804953061193481451918217412731409759930588445568} a^{7} - \frac{6516419452107782613794708145759838599307393352519862264230505269981674668749503520301631744878474035685229}{40818459613530261892366580391995416301760443469444749311041804953061193481451918217412731409759930588445568} a^{6} + \frac{6355354304543490808323541316479875197373067214306594066631476649190305191962573213732465125908186929044939}{40818459613530261892366580391995416301760443469444749311041804953061193481451918217412731409759930588445568} a^{5} - \frac{19661289634073768079603172540459597374866827045606561373392804466011876454273102008177976507294330093971119}{40818459613530261892366580391995416301760443469444749311041804953061193481451918217412731409759930588445568} a^{4} - \frac{7393165265638561381916214429626592690021919115840532356670706361089365436120599461462909194083319674187241}{40818459613530261892366580391995416301760443469444749311041804953061193481451918217412731409759930588445568} a^{3} + \frac{29113544224400095889998675166703128330265746749879390977462195522346592208974626858511620634399835240123933}{81636919227060523784733160783990832603520886938889498622083609906122386962903836434825462819519861176891136} a^{2} - \frac{5589252327375152012632433373116149751049843651327960131893307682911390526591030489825536915746428474441317}{81636919227060523784733160783990832603520886938889498622083609906122386962903836434825462819519861176891136} a + \frac{22255329852239956243670323942343906748389153349994151701523629905753671335959132977555445329228703879893853}{81636919227060523784733160783990832603520886938889498622083609906122386962903836434825462819519861176891136}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.1090397.1, 10.0.246071287.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ $20$ $20$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ $20$

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
7Data not computed
$11$11.10.9.5$x^{10} - 8019$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
17Data not computed