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Properties

Label 20.0.16468295313...0000.9
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $144.82$
Ramified primes $2, 5, 7, 11$
Class number $20539360$ (GRH)
Class group $[4, 5134840]$ (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![292007118139, -53148809186, 116125201467, -16231085948, 23097010519, -2733365734, 3107548543, -316020408, 307055859, -27106178, 23492371, -1807130, 1404483, -91076, 65679, -3662, 2359, -96, 59, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 59*x^18 - 96*x^17 + 2359*x^16 - 3662*x^15 + 65679*x^14 - 91076*x^13 + 1404483*x^12 - 1807130*x^11 + 23492371*x^10 - 27106178*x^9 + 307055859*x^8 - 316020408*x^7 + 3107548543*x^6 - 2733365734*x^5 + 23097010519*x^4 - 16231085948*x^3 + 116125201467*x^2 - 53148809186*x + 292007118139)
 
gp: K = bnfinit(x^20 - 2*x^19 + 59*x^18 - 96*x^17 + 2359*x^16 - 3662*x^15 + 65679*x^14 - 91076*x^13 + 1404483*x^12 - 1807130*x^11 + 23492371*x^10 - 27106178*x^9 + 307055859*x^8 - 316020408*x^7 + 3107548543*x^6 - 2733365734*x^5 + 23097010519*x^4 - 16231085948*x^3 + 116125201467*x^2 - 53148809186*x + 292007118139, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 59 x^{18} - 96 x^{17} + 2359 x^{16} - 3662 x^{15} + 65679 x^{14} - 91076 x^{13} + 1404483 x^{12} - 1807130 x^{11} + 23492371 x^{10} - 27106178 x^{9} + 307055859 x^{8} - 316020408 x^{7} + 3107548543 x^{6} - 2733365734 x^{5} + 23097010519 x^{4} - 16231085948 x^{3} + 116125201467 x^{2} - 53148809186 x + 292007118139 \)

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:  \(16468295313503070686136120314429440000000000=2^{30}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $144.82$
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:  $2, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3080=2^{3}\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3080}(1,·)$, $\chi_{3080}(1539,·)$, $\chi_{3080}(699,·)$, $\chi_{3080}(321,·)$, $\chi_{3080}(841,·)$, $\chi_{3080}(139,·)$, $\chi_{3080}(2001,·)$, $\chi_{3080}(1219,·)$, $\chi_{3080}(601,·)$, $\chi_{3080}(1499,·)$, $\chi_{3080}(2619,·)$, $\chi_{3080}(1681,·)$, $\chi_{3080}(41,·)$, $\chi_{3080}(939,·)$, $\chi_{3080}(2659,·)$, $\chi_{3080}(1961,·)$, $\chi_{3080}(1161,·)$, $\chi_{3080}(1401,·)$, $\chi_{3080}(2939,·)$, $\chi_{3080}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{23} a^{11} + \frac{7}{23} a^{10} - \frac{6}{23} a^{9} - \frac{1}{23} a^{8} + \frac{1}{23} a^{7} + \frac{11}{23} a^{6} + \frac{1}{23} a^{5} + \frac{5}{23} a^{4} - \frac{1}{23} a^{3} + \frac{8}{23} a^{2} + \frac{3}{23} a - \frac{8}{23}$, $\frac{1}{23} a^{12} - \frac{9}{23} a^{10} - \frac{5}{23} a^{9} + \frac{8}{23} a^{8} + \frac{4}{23} a^{7} - \frac{7}{23} a^{6} - \frac{2}{23} a^{5} + \frac{10}{23} a^{4} - \frac{8}{23} a^{3} - \frac{7}{23} a^{2} - \frac{6}{23} a + \frac{10}{23}$, $\frac{1}{23} a^{13} - \frac{11}{23} a^{10} - \frac{5}{23} a^{8} + \frac{2}{23} a^{7} + \frac{5}{23} a^{6} - \frac{4}{23} a^{5} - \frac{9}{23} a^{4} + \frac{7}{23} a^{3} - \frac{3}{23} a^{2} - \frac{9}{23} a - \frac{3}{23}$, $\frac{1}{23} a^{14} + \frac{8}{23} a^{10} - \frac{2}{23} a^{9} - \frac{9}{23} a^{8} - \frac{7}{23} a^{7} + \frac{2}{23} a^{6} + \frac{2}{23} a^{5} - \frac{7}{23} a^{4} + \frac{9}{23} a^{3} + \frac{10}{23} a^{2} + \frac{7}{23} a + \frac{4}{23}$, $\frac{1}{23} a^{15} + \frac{11}{23} a^{10} - \frac{7}{23} a^{9} + \frac{1}{23} a^{8} - \frac{6}{23} a^{7} + \frac{6}{23} a^{6} + \frac{8}{23} a^{5} - \frac{8}{23} a^{4} - \frac{5}{23} a^{3} - \frac{11}{23} a^{2} + \frac{3}{23} a - \frac{5}{23}$, $\frac{1}{23} a^{16} + \frac{8}{23} a^{10} - \frac{2}{23} a^{9} + \frac{5}{23} a^{8} - \frac{5}{23} a^{7} + \frac{2}{23} a^{6} + \frac{4}{23} a^{5} + \frac{9}{23} a^{4} + \frac{7}{23} a^{2} + \frac{8}{23} a - \frac{4}{23}$, $\frac{1}{5543} a^{17} - \frac{88}{5543} a^{16} - \frac{3}{5543} a^{15} + \frac{99}{5543} a^{14} - \frac{107}{5543} a^{13} - \frac{89}{5543} a^{12} + \frac{60}{5543} a^{11} + \frac{141}{5543} a^{10} - \frac{1864}{5543} a^{9} - \frac{602}{5543} a^{8} - \frac{659}{5543} a^{7} + \frac{1956}{5543} a^{6} - \frac{1006}{5543} a^{5} + \frac{2667}{5543} a^{4} - \frac{1844}{5543} a^{3} + \frac{809}{5543} a^{2} + \frac{410}{5543} a + \frac{2354}{5543}$, $\frac{1}{127489} a^{18} - \frac{8}{127489} a^{17} - \frac{1259}{127489} a^{16} - \frac{382}{127489} a^{15} + \frac{342}{127489} a^{14} + \frac{1473}{127489} a^{13} - \frac{1276}{127489} a^{12} + \frac{1326}{127489} a^{11} - \frac{45773}{127489} a^{10} + \frac{3313}{127489} a^{9} - \frac{36769}{127489} a^{8} - \frac{41606}{127489} a^{7} - \frac{17564}{127489} a^{6} - \frac{32987}{127489} a^{5} - \frac{49487}{127489} a^{4} + \frac{20543}{127489} a^{3} - \frac{44043}{127489} a^{2} - \frac{36182}{127489} a - \frac{6649}{127489}$, $\frac{1}{6425580373907526411962354327144184983189091838451178164011731429562259} a^{19} + \frac{8931080063652798329517263443597814749600093899784514542881509375}{6425580373907526411962354327144184983189091838451178164011731429562259} a^{18} + \frac{240815976284250480527189237912569112544921321034243137792778437644}{6425580373907526411962354327144184983189091838451178164011731429562259} a^{17} - \frac{33419808549107156691502999639374736278616800279510084709675213083600}{6425580373907526411962354327144184983189091838451178164011731429562259} a^{16} - \frac{120032544190245409360564043488489612514530087896175743595351581534480}{6425580373907526411962354327144184983189091838451178164011731429562259} a^{15} + \frac{21591992169115886629638898939725657924817024789111558961868597650610}{6425580373907526411962354327144184983189091838451178164011731429562259} a^{14} + \frac{74809003513663224227837011336129887967908708555232535638344867601857}{6425580373907526411962354327144184983189091838451178164011731429562259} a^{13} - \frac{77478924288087879691249580459972844804915291987401971435764298434175}{6425580373907526411962354327144184983189091838451178164011731429562259} a^{12} - \frac{35382802734609849723163980516684803425303568151722893917974020236497}{6425580373907526411962354327144184983189091838451178164011731429562259} a^{11} + \frac{2699926550926542352752389997439701507836337921465140492619493669869752}{6425580373907526411962354327144184983189091838451178164011731429562259} a^{10} + \frac{633579779154927093722153023459545363052105291563795137918701379780747}{6425580373907526411962354327144184983189091838451178164011731429562259} a^{9} - \frac{1506629991236052811259808808564048247065268103576070111790481878872274}{6425580373907526411962354327144184983189091838451178164011731429562259} a^{8} + \frac{2777439953395250138775991815055172975174835269430208824609183283455605}{6425580373907526411962354327144184983189091838451178164011731429562259} a^{7} - \frac{1456955260663241851014306694041108022397337412277537154605699675370264}{6425580373907526411962354327144184983189091838451178164011731429562259} a^{6} - \frac{1059659664121847537198442777838906913577797975758251529968654999604236}{6425580373907526411962354327144184983189091838451178164011731429562259} a^{5} + \frac{2901869539080811752278961860555248118273225266570483257167499821922644}{6425580373907526411962354327144184983189091838451178164011731429562259} a^{4} - \frac{2874352722900175968155467720220408789746818076273838784994662835964674}{6425580373907526411962354327144184983189091838451178164011731429562259} a^{3} - \frac{876128758732074496524151062137383923175112972766850987874676937301234}{6425580373907526411962354327144184983189091838451178164011731429562259} a^{2} - \frac{797315681772250194054368083287877924899705201322294566687493040349110}{6425580373907526411962354327144184983189091838451178164011731429562259} a - \frac{1411550717196740699437198882513188112983724372356441031133240800305216}{6425580373907526411962354327144184983189091838451178164011731429562259}$

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

Class group and class number

$C_{4}\times C_{5134840}$, which has order $20539360$ (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:  \( 1415140.1624943546 \) (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

\(\Q(\sqrt{77}) \), \(\Q(\sqrt{-770}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-10}, \sqrt{77})\), \(\Q(\zeta_{11})^+\), 10.10.39630026842637.1, 10.0.4058114748686028800000.1, 10.0.21950349414400000.4

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R R R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$