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Properties

Label 20.0.23695741335...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 5^{32}\cdot 7^{10}$
Root discriminant $233.74$
Ramified primes $2, 5, 7$
Class number $138732544$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 2, 270962]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![309599949457, 55548895040, 171747552260, 37655714520, 43045152610, 7340687380, 6379299140, 510835300, 521842005, 9872500, 41759034, -62500, 3104840, -600, 157350, -524, 5050, -20, 100, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 100*x^18 - 20*x^17 + 5050*x^16 - 524*x^15 + 157350*x^14 - 600*x^13 + 3104840*x^12 - 62500*x^11 + 41759034*x^10 + 9872500*x^9 + 521842005*x^8 + 510835300*x^7 + 6379299140*x^6 + 7340687380*x^5 + 43045152610*x^4 + 37655714520*x^3 + 171747552260*x^2 + 55548895040*x + 309599949457)
 
gp: K = bnfinit(x^20 + 100*x^18 - 20*x^17 + 5050*x^16 - 524*x^15 + 157350*x^14 - 600*x^13 + 3104840*x^12 - 62500*x^11 + 41759034*x^10 + 9872500*x^9 + 521842005*x^8 + 510835300*x^7 + 6379299140*x^6 + 7340687380*x^5 + 43045152610*x^4 + 37655714520*x^3 + 171747552260*x^2 + 55548895040*x + 309599949457, 1)
 

Normalized defining polynomial

\( x^{20} + 100 x^{18} - 20 x^{17} + 5050 x^{16} - 524 x^{15} + 157350 x^{14} - 600 x^{13} + 3104840 x^{12} - 62500 x^{11} + 41759034 x^{10} + 9872500 x^{9} + 521842005 x^{8} + 510835300 x^{7} + 6379299140 x^{6} + 7340687380 x^{5} + 43045152610 x^{4} + 37655714520 x^{3} + 171747552260 x^{2} + 55548895040 x + 309599949457 \)

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:  \(236957413356339200000000000000000000000000000000=2^{55}\cdot 5^{32}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $233.74$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2800=2^{4}\cdot 5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{2800}(1,·)$, $\chi_{2800}(1861,·)$, $\chi_{2800}(2241,·)$, $\chi_{2800}(841,·)$, $\chi_{2800}(2701,·)$, $\chi_{2800}(461,·)$, $\chi_{2800}(1681,·)$, $\chi_{2800}(1301,·)$, $\chi_{2800}(281,·)$, $\chi_{2800}(2521,·)$, $\chi_{2800}(2141,·)$, $\chi_{2800}(1121,·)$, $\chi_{2800}(741,·)$, $\chi_{2800}(1961,·)$, $\chi_{2800}(1581,·)$, $\chi_{2800}(561,·)$, $\chi_{2800}(2421,·)$, $\chi_{2800}(1401,·)$, $\chi_{2800}(1021,·)$, $\chi_{2800}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{2}{7} a^{5} + \frac{3}{7} a^{4} - \frac{1}{7} a^{3} - \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} + \frac{3}{7} a^{4} + \frac{1}{7} a^{3} - \frac{1}{7} a^{2} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{10} + \frac{3}{7} a^{7} + \frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a^{2} + \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{11} - \frac{2}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2} + \frac{2}{7}$, $\frac{1}{7} a^{12} - \frac{2}{7} a^{6} + \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{2}{7} a^{7} + \frac{1}{7} a$, $\frac{1}{7} a^{14} + \frac{2}{7} a^{7} - \frac{3}{7} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{3} + \frac{1}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{15} - \frac{2}{7} a^{7} - \frac{3}{7} a^{6} + \frac{2}{7} a^{5} - \frac{1}{7} a^{4} + \frac{3}{7} a^{3} - \frac{2}{7} a^{2} + \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{49} a^{16} + \frac{2}{49} a^{15} + \frac{1}{49} a^{14} - \frac{3}{49} a^{13} + \frac{3}{49} a^{12} - \frac{3}{49} a^{11} + \frac{2}{49} a^{10} + \frac{3}{49} a^{9} - \frac{3}{49} a^{8} + \frac{23}{49} a^{7} + \frac{11}{49} a^{6} - \frac{11}{49} a^{5} - \frac{9}{49} a^{4} - \frac{8}{49} a^{3} - \frac{6}{49} a^{2} + \frac{13}{49} a + \frac{16}{49}$, $\frac{1}{343} a^{17} - \frac{2}{343} a^{16} - \frac{1}{49} a^{15} + \frac{3}{49} a^{14} + \frac{1}{343} a^{13} - \frac{22}{343} a^{12} + \frac{2}{49} a^{11} + \frac{2}{343} a^{10} - \frac{8}{343} a^{9} - \frac{3}{49} a^{8} + \frac{108}{343} a^{7} - \frac{20}{343} a^{6} + \frac{11}{49} a^{5} - \frac{3}{49} a^{4} + \frac{68}{343} a^{3} - \frac{5}{343} a^{2} + \frac{48}{343} a - \frac{120}{343}$, $\frac{1}{2924417461233624207451298101543} a^{18} + \frac{3130298566656549010022855552}{2924417461233624207451298101543} a^{17} + \frac{12031555644275012798089614787}{2924417461233624207451298101543} a^{16} + \frac{20073835473570955521001833440}{417773923033374886778756871649} a^{15} + \frac{97389581126030477985574460266}{2924417461233624207451298101543} a^{14} - \frac{195099028568419886244284631683}{2924417461233624207451298101543} a^{13} - \frac{64985844997354922438917360590}{2924417461233624207451298101543} a^{12} + \frac{35069262224101511294401382095}{2924417461233624207451298101543} a^{11} - \frac{72352373962827967935372099888}{2924417461233624207451298101543} a^{10} + \frac{201344743421155828096994482883}{2924417461233624207451298101543} a^{9} - \frac{100045184053644654331256049423}{2924417461233624207451298101543} a^{8} + \frac{770454028181592560889894281315}{2924417461233624207451298101543} a^{7} - \frac{742114469604296741325907523641}{2924417461233624207451298101543} a^{6} + \frac{119837270508537704054043399269}{417773923033374886778756871649} a^{5} + \frac{528147520136492124998281331429}{2924417461233624207451298101543} a^{4} + \frac{1074905878259605767940519206711}{2924417461233624207451298101543} a^{3} - \frac{1420024088484474391033899454401}{2924417461233624207451298101543} a^{2} + \frac{83246909065257029698025502745}{417773923033374886778756871649} a - \frac{122986456053475106659066833283}{2924417461233624207451298101543}$, $\frac{1}{6260057065386162291159402195995159578006030570589855747851966001} a^{19} + \frac{484961395402777089309068851427409}{6260057065386162291159402195995159578006030570589855747851966001} a^{18} + \frac{7304149763802179622946291968671653410530411286144313073987834}{6260057065386162291159402195995159578006030570589855747851966001} a^{17} - \frac{62095437560198864490137087632412793207926915035457025428293972}{6260057065386162291159402195995159578006030570589855747851966001} a^{16} + \frac{229197582868714360026589981058137748866825246287647685443825320}{6260057065386162291159402195995159578006030570589855747851966001} a^{15} - \frac{79323961349580559797272167563100419139587273885563012632909309}{6260057065386162291159402195995159578006030570589855747851966001} a^{14} + \frac{433701826082726147761745997950529556404335693580232765633885089}{6260057065386162291159402195995159578006030570589855747851966001} a^{13} + \frac{50752087275843588825816523678158800857903282805267689621363274}{894293866483737470165628885142165654000861510084265106835995143} a^{12} + \frac{13807356848866686505961595682120070898369026900608400120949931}{6260057065386162291159402195995159578006030570589855747851966001} a^{11} - \frac{369148001059262193079685723925424728452999238892021545396752583}{6260057065386162291159402195995159578006030570589855747851966001} a^{10} + \frac{172780939505714665251619478258789584997518519657304156036173123}{6260057065386162291159402195995159578006030570589855747851966001} a^{9} + \frac{354039664062518619654953588046645678967972680148561398971512707}{6260057065386162291159402195995159578006030570589855747851966001} a^{8} + \frac{425861461976169017454668534168344083412728136599137515450149785}{6260057065386162291159402195995159578006030570589855747851966001} a^{7} - \frac{2985754337885861110172623471870080545697358634797925478756497482}{6260057065386162291159402195995159578006030570589855747851966001} a^{6} + \frac{135936853472880078231400800249031932076967392901751777259590814}{6260057065386162291159402195995159578006030570589855747851966001} a^{5} + \frac{1854146612042548258060723349997397898748649242527969386148469859}{6260057065386162291159402195995159578006030570589855747851966001} a^{4} + \frac{17865111801751711088500778680462128181865144152981791669187772}{127756266640533924309375555020309379142980215726323586690856449} a^{3} - \frac{519232540817554114491109297725715058071793888278581314925977337}{6260057065386162291159402195995159578006030570589855747851966001} a^{2} - \frac{2417961932012106452105076391420412366199809705700265256635030072}{6260057065386162291159402195995159578006030570589855747851966001} a + \frac{3056351190287808431610837589083332903824295199195876865004355269}{6260057065386162291159402195995159578006030570589855747851966001}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{270962}$, which has order $138732544$ (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:  \( 42294001.73672045 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

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

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.100352.5, 5.5.390625.1, 10.10.5000000000000000.1

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 R $20$ R R $20$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ $20$

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
2Data not computed
5Data not computed
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$