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Properties

Label 20.0.35635746108...1401.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 7^{10}\cdot 271^{10}$
Root discriminant $75.44$
Ramified primes $3, 7, 271$
Class number $10092$ (GRH)
Class group $[58, 174]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![541725625, -482420925, 364129143, -196740684, 92157240, -16211013, 2786196, -256473, -59897, -121860, -50276, 53466, -18345, 6402, -1314, 282, -56, -30, 17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 17*x^18 - 30*x^17 - 56*x^16 + 282*x^15 - 1314*x^14 + 6402*x^13 - 18345*x^12 + 53466*x^11 - 50276*x^10 - 121860*x^9 - 59897*x^8 - 256473*x^7 + 2786196*x^6 - 16211013*x^5 + 92157240*x^4 - 196740684*x^3 + 364129143*x^2 - 482420925*x + 541725625)
 
gp: K = bnfinit(x^20 - 6*x^19 + 17*x^18 - 30*x^17 - 56*x^16 + 282*x^15 - 1314*x^14 + 6402*x^13 - 18345*x^12 + 53466*x^11 - 50276*x^10 - 121860*x^9 - 59897*x^8 - 256473*x^7 + 2786196*x^6 - 16211013*x^5 + 92157240*x^4 - 196740684*x^3 + 364129143*x^2 - 482420925*x + 541725625, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 17 x^{18} - 30 x^{17} - 56 x^{16} + 282 x^{15} - 1314 x^{14} + 6402 x^{13} - 18345 x^{12} + 53466 x^{11} - 50276 x^{10} - 121860 x^{9} - 59897 x^{8} - 256473 x^{7} + 2786196 x^{6} - 16211013 x^{5} + 92157240 x^{4} - 196740684 x^{3} + 364129143 x^{2} - 482420925 x + 541725625 \)

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:  \(35635746108202593567757236917916961401=3^{10}\cdot 7^{10}\cdot 271^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.44$
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:  $3, 7, 271$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7} a^{11} + \frac{1}{7} a^{10} + \frac{2}{7} a^{9} + \frac{2}{7} a^{8} + \frac{3}{7} a^{7} + \frac{3}{7} a^{6} + \frac{2}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{10} + \frac{1}{7} a^{8} - \frac{1}{7} a^{6} - \frac{1}{7} a^{4} - \frac{1}{7} a^{2}$, $\frac{1}{133} a^{13} + \frac{9}{133} a^{12} - \frac{8}{133} a^{11} - \frac{7}{19} a^{10} + \frac{18}{133} a^{9} + \frac{5}{133} a^{8} - \frac{9}{19} a^{7} + \frac{20}{133} a^{6} - \frac{12}{133} a^{5} - \frac{41}{133} a^{4} - \frac{17}{133} a^{3} + \frac{3}{133} a^{2} - \frac{8}{19} a$, $\frac{1}{133} a^{14} + \frac{6}{133} a^{12} + \frac{4}{133} a^{11} + \frac{3}{133} a^{10} - \frac{62}{133} a^{9} - \frac{51}{133} a^{8} - \frac{2}{133} a^{7} + \frac{55}{133} a^{6} + \frac{29}{133} a^{5} - \frac{47}{133} a^{4} + \frac{4}{133} a^{3} - \frac{64}{133} a^{2} - \frac{4}{19} a$, $\frac{1}{4655} a^{15} - \frac{16}{4655} a^{14} + \frac{9}{4655} a^{13} - \frac{274}{4655} a^{12} + \frac{276}{4655} a^{11} + \frac{327}{665} a^{10} + \frac{1584}{4655} a^{9} + \frac{295}{931} a^{8} - \frac{1413}{4655} a^{7} - \frac{2159}{4655} a^{6} + \frac{9}{95} a^{5} + \frac{1963}{4655} a^{4} - \frac{37}{95} a^{3} - \frac{326}{665} a^{2} - \frac{16}{95} a$, $\frac{1}{4655} a^{16} - \frac{2}{4655} a^{14} + \frac{2}{931} a^{13} - \frac{48}{4655} a^{12} - \frac{17}{931} a^{11} - \frac{1167}{4655} a^{10} + \frac{849}{4655} a^{9} - \frac{1578}{4655} a^{8} + \frac{1833}{4655} a^{7} - \frac{1868}{4655} a^{6} + \frac{1144}{4655} a^{5} - \frac{458}{931} a^{4} - \frac{60}{133} a^{3} - \frac{3}{665} a^{2} + \frac{14}{95} a$, $\frac{1}{4655} a^{17} + \frac{13}{4655} a^{14} + \frac{1}{931} a^{13} - \frac{108}{4655} a^{12} - \frac{18}{931} a^{11} - \frac{173}{4655} a^{10} + \frac{276}{931} a^{9} - \frac{8}{245} a^{8} - \frac{319}{4655} a^{7} + \frac{1446}{4655} a^{6} + \frac{517}{4655} a^{5} + \frac{4}{245} a^{4} + \frac{174}{665} a^{3} - \frac{99}{665} a^{2} + \frac{3}{95} a$, $\frac{1}{8158237807775408605} a^{18} - \frac{72522783994511}{1631647561555081721} a^{17} + \frac{550715051593017}{8158237807775408605} a^{16} + \frac{151632301165322}{8158237807775408605} a^{15} - \frac{3766100453191249}{1165462543967915515} a^{14} + \frac{20244095039920383}{8158237807775408605} a^{13} + \frac{249703452168466788}{8158237807775408605} a^{12} - \frac{311811784126088424}{8158237807775408605} a^{11} - \frac{3101615353577491938}{8158237807775408605} a^{10} + \frac{425346919805014891}{1165462543967915515} a^{9} + \frac{647211535793208373}{1631647561555081721} a^{8} + \frac{8327544308852702}{95979268326769513} a^{7} - \frac{85414855856179788}{1631647561555081721} a^{6} + \frac{529945399566354384}{1165462543967915515} a^{5} - \frac{80014092116560059}{233092508793583103} a^{4} - \frac{978919041920633}{33298929827654729} a^{3} - \frac{37226734417720307}{166494649138273645} a^{2} + \frac{211147569738278}{1251839467205065} a - \frac{947143819820}{13177257549527}$, $\frac{1}{33300064561385482837743128991355497207929975} a^{19} + \frac{375228333172572815027849}{33300064561385482837743128991355497207929975} a^{18} - \frac{3272797054639318237096779309500749343573}{33300064561385482837743128991355497207929975} a^{17} + \frac{359959470460931746603235441496042277958}{6660012912277096567548625798271099441585995} a^{16} + \frac{69593060453886490043835803132402722292}{4757152080197926119677589855907928172561425} a^{15} + \frac{991410194473163267150842821051448920653}{1148278088323637339232521689357086110618275} a^{14} - \frac{33725691375027369116607084558723424910189}{33300064561385482837743128991355497207929975} a^{13} + \frac{12263967567508702284099942424730935259738}{1752634976915025412512796262702920905680525} a^{12} + \frac{3370224025728169780676747249449917830743}{1332002582455419313509725159654219888317199} a^{11} - \frac{48673819796615184948691158871463382470913}{164039726903376762747503098479583730088325} a^{10} - \frac{372593783263382524254001215725688962741341}{33300064561385482837743128991355497207929975} a^{9} - \frac{1199412093175020125583840063706146975367937}{6660012912277096567548625798271099441585995} a^{8} - \frac{15821085706181402739160439042610841630873052}{33300064561385482837743128991355497207929975} a^{7} - \frac{12617109588081139843438441333105910651766}{39976067900822908568719242486621245147575} a^{6} + \frac{104283309332559065247417683680656095541164}{679593154313989445668227122272561167508775} a^{5} + \frac{238105546040575355197266354170505259580283}{679593154313989445668227122272561167508775} a^{4} + \frac{47999644561314655851460448276795690210}{3883389453222796832389869270128920957193} a^{3} - \frac{5663240205950519078117325459674066533194}{13869248047224274401392390250460431989975} a^{2} - \frac{1807689259990566743521251748651675282396}{5109722964766837937355091144906474943675} a - \frac{76343720553198033549853304192663880}{1536758786396041484918824404483150359}$

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

Class group and class number

$C_{58}\times C_{174}$, which has order $10092$ (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:  \( -\frac{227002405619765552532824}{42960463921711350908262596858225} a^{19} + \frac{56329089473098914311297}{2527086113041844171074270403425} a^{18} - \frac{1775019398771613940912408}{42960463921711350908262596858225} a^{17} + \frac{17308074879947558492179}{8592092784342270181652519371645} a^{16} + \frac{2611523155953604170428187}{6137209131673050129751799551175} a^{15} - \frac{16942992925497691105757518}{42960463921711350908262596858225} a^{14} + \frac{155875202678428069089698596}{42960463921711350908262596858225} a^{13} - \frac{872091234553851828579270773}{42960463921711350908262596858225} a^{12} + \frac{415784073135469142977646758}{8592092784342270181652519371645} a^{11} - \frac{2083400244464798955298444}{19000647466480031361460679725} a^{10} - \frac{7177648829836324721495546461}{42960463921711350908262596858225} a^{9} + \frac{5299556183691243077951842884}{8592092784342270181652519371645} a^{8} + \frac{86564836802462463047405632883}{42960463921711350908262596858225} a^{7} - \frac{1363807187573929475487690634}{6137209131673050129751799551175} a^{6} - \frac{8434774917975810525612764951}{876744161667578589964542793025} a^{5} + \frac{59743420119315602622227463548}{876744161667578589964542793025} a^{4} - \frac{1205767378303689843078506639}{5009966638100449085511673103} a^{3} + \frac{15389541296002707544936801557}{125249165952511227137791827575} a^{2} - \frac{94562633017542943175553438}{387768315642449619621646525} a + \frac{1184294886764986908035361}{1982574846893727378516689} \) (order $6$)
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:  \( 80801816.0664 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{1897}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-5691}) \), \(\Q(\sqrt{-3}, \sqrt{1897})\), 5.5.3598609.1 x5, 10.10.24566124836069257.1, 10.0.3146846776576083.1 x5, 10.0.5969568335164829451.1 x5

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

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
271Data not computed