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Properties

Label 20.4.38166916322...8125.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{31}\cdot 31^{10}$
Root discriminant $67.47$
Ramified primes $5, 31$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49350856976, -12345750840, 18277709420, 2198708690, -10721050195, -255897903, 1894042635, -217085270, -99979485, 53786260, -12799606, -575235, 839990, -185365, 24500, -1282, -1075, 390, -25, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 25*x^18 + 390*x^17 - 1075*x^16 - 1282*x^15 + 24500*x^14 - 185365*x^13 + 839990*x^12 - 575235*x^11 - 12799606*x^10 + 53786260*x^9 - 99979485*x^8 - 217085270*x^7 + 1894042635*x^6 - 255897903*x^5 - 10721050195*x^4 + 2198708690*x^3 + 18277709420*x^2 - 12345750840*x + 49350856976)
 
gp: K = bnfinit(x^20 - 5*x^19 - 25*x^18 + 390*x^17 - 1075*x^16 - 1282*x^15 + 24500*x^14 - 185365*x^13 + 839990*x^12 - 575235*x^11 - 12799606*x^10 + 53786260*x^9 - 99979485*x^8 - 217085270*x^7 + 1894042635*x^6 - 255897903*x^5 - 10721050195*x^4 + 2198708690*x^3 + 18277709420*x^2 - 12345750840*x + 49350856976, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 25 x^{18} + 390 x^{17} - 1075 x^{16} - 1282 x^{15} + 24500 x^{14} - 185365 x^{13} + 839990 x^{12} - 575235 x^{11} - 12799606 x^{10} + 53786260 x^{9} - 99979485 x^{8} - 217085270 x^{7} + 1894042635 x^{6} - 255897903 x^{5} - 10721050195 x^{4} + 2198708690 x^{3} + 18277709420 x^{2} - 12345750840 x + 49350856976 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3816691632293169386684894561767578125=5^{31}\cdot 31^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.47$
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:  $5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{18} + \frac{1}{12} a^{17} - \frac{1}{24} a^{16} - \frac{5}{24} a^{15} - \frac{1}{6} a^{14} - \frac{1}{6} a^{13} + \frac{7}{24} a^{11} + \frac{1}{8} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{7} + \frac{11}{24} a^{6} + \frac{11}{24} a^{5} + \frac{5}{12} a^{4} + \frac{3}{8} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{99665014702776569912528137011145849096262456139761147742875723233607230529591093926994797236349416} a^{19} - \frac{77329570452638691683746045537996589967000977479003233285943827186035626571573748441353775402161}{24916253675694142478132034252786462274065614034940286935718930808401807632397773481748699309087354} a^{18} + \frac{2541981338282297901377494484413718546984890101894569923767110684488565305958128699752898498400867}{33221671567592189970842712337048616365420818713253715914291907744535743509863697975664932412116472} a^{17} + \frac{5321510040987033219010120843525189386434639119489545256850055990865940663408261541727068741060503}{99665014702776569912528137011145849096262456139761147742875723233607230529591093926994797236349416} a^{16} + \frac{7548620725799380217171903891356393481553962270366826036238096806334159127360700939165769771738091}{49832507351388284956264068505572924548131228069880573871437861616803615264795546963497398618174708} a^{15} - \frac{15687329850631981119980035399804281477873764404421338916507810934569397565279306845247775227881}{16610835783796094985421356168524308182710409356626857957145953872267871754931848987832466206058236} a^{14} + \frac{9352330361874417129865491293059251039410603142016695175137399798742787743026733336592898612672515}{49832507351388284956264068505572924548131228069880573871437861616803615264795546963497398618174708} a^{13} + \frac{15814643956309870805176761906993218221727405708754389695122816191266430921552799296750249997094897}{99665014702776569912528137011145849096262456139761147742875723233607230529591093926994797236349416} a^{12} + \frac{21377971368673961529109500556180912633991377942359083884345332780382703190662053422923912673861731}{99665014702776569912528137011145849096262456139761147742875723233607230529591093926994797236349416} a^{11} + \frac{4761809945844259260660474315924222323690269233776435114678956684715833162987576604431117682839313}{49832507351388284956264068505572924548131228069880573871437861616803615264795546963497398618174708} a^{10} + \frac{23946107655655648163692034676804399426458926436334576658608777272738949295130817176236715425124149}{49832507351388284956264068505572924548131228069880573871437861616803615264795546963497398618174708} a^{9} + \frac{15786702354442770554080584439854272149191621368448894735050024505556121651250209993142253814592151}{49832507351388284956264068505572924548131228069880573871437861616803615264795546963497398618174708} a^{8} - \frac{6031679119066955304277505852615928592277219650880342201183269914496070596335573709229512087176325}{33221671567592189970842712337048616365420818713253715914291907744535743509863697975664932412116472} a^{7} + \frac{9097732784471604442895065357448300781445734100750625837475704616254512963492932538328768504283}{32222765826956537314105443585886145844249096715086048413474207317687433084251889404136694871112} a^{6} + \frac{11331987917230991500769813682542632122688537886111879544866028407978739429556654712277398076420701}{24916253675694142478132034252786462274065614034940286935718930808401807632397773481748699309087354} a^{5} - \frac{29779860481459048022330796655108127747510686765082464471071897226833617773998861487013822963034229}{99665014702776569912528137011145849096262456139761147742875723233607230529591093926994797236349416} a^{4} - \frac{1726056980173391629879549649354756585762484187865034090902470291826433280546077667482662837978553}{8305417891898047492710678084262154091355204678313428978572976936133935877465924493916233103029118} a^{3} + \frac{10362483766601805756858773511542044827580420237610732304511971766795109244103665029288547835679603}{24916253675694142478132034252786462274065614034940286935718930808401807632397773481748699309087354} a^{2} + \frac{1884583534534815466301102465901234273162974294464820334713226373566829935939363298106544209686987}{12458126837847071239066017126393231137032807017470143467859465404200903816198886740874349654543677} a - \frac{4787664214237150069521882674044916819476504548104938641722116835938756337156949993740829860484233}{12458126837847071239066017126393231137032807017470143467859465404200903816198886740874349654543677}$

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:  $11$
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

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.120125.1, 5.1.78125.1, 10.2.30517578125.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$31$31.10.5.2$x^{10} - 923521 x^{2} + 286291510$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
31.10.5.2$x^{10} - 923521 x^{2} + 286291510$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$