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Properties

Label 20.0.13350881126...0625.3
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 31^{10}$
Root discriminant $57.05$
Ramified primes $3, 5, 7, 31$
Class number $3872$ (GRH)
Class group $[2, 22, 88]$ (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![562195, -443775, 269196, 467764, -409139, -237809, 1187558, -1790508, 1781990, -1378339, 853479, -435736, 184128, -63680, 18294, -4180, 905, -192, 53, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 53*x^18 - 192*x^17 + 905*x^16 - 4180*x^15 + 18294*x^14 - 63680*x^13 + 184128*x^12 - 435736*x^11 + 853479*x^10 - 1378339*x^9 + 1781990*x^8 - 1790508*x^7 + 1187558*x^6 - 237809*x^5 - 409139*x^4 + 467764*x^3 + 269196*x^2 - 443775*x + 562195)
 
gp: K = bnfinit(x^20 - 10*x^19 + 53*x^18 - 192*x^17 + 905*x^16 - 4180*x^15 + 18294*x^14 - 63680*x^13 + 184128*x^12 - 435736*x^11 + 853479*x^10 - 1378339*x^9 + 1781990*x^8 - 1790508*x^7 + 1187558*x^6 - 237809*x^5 - 409139*x^4 + 467764*x^3 + 269196*x^2 - 443775*x + 562195, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 53 x^{18} - 192 x^{17} + 905 x^{16} - 4180 x^{15} + 18294 x^{14} - 63680 x^{13} + 184128 x^{12} - 435736 x^{11} + 853479 x^{10} - 1378339 x^{9} + 1781990 x^{8} - 1790508 x^{7} + 1187558 x^{6} - 237809 x^{5} - 409139 x^{4} + 467764 x^{3} + 269196 x^{2} - 443775 x + 562195 \)

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:  \(133508811261782548210805849619140625=3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 31^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.05$
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:  $3, 5, 7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{8} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{9} + \frac{1}{5} a^{5}$, $\frac{1}{75} a^{14} - \frac{7}{75} a^{13} + \frac{1}{25} a^{12} - \frac{2}{75} a^{11} + \frac{1}{15} a^{10} - \frac{12}{25} a^{9} + \frac{19}{75} a^{8} + \frac{29}{75} a^{7} - \frac{8}{75} a^{6} + \frac{1}{25} a^{5} + \frac{23}{75} a^{4} - \frac{17}{75} a^{3} + \frac{4}{25} a^{2} - \frac{1}{3} a - \frac{7}{15}$, $\frac{1}{375} a^{15} - \frac{1}{375} a^{13} - \frac{11}{375} a^{12} + \frac{12}{125} a^{11} - \frac{31}{375} a^{10} - \frac{23}{375} a^{9} - \frac{1}{125} a^{8} - \frac{3}{25} a^{7} - \frac{143}{375} a^{6} + \frac{14}{375} a^{5} - \frac{37}{125} a^{4} + \frac{13}{375} a^{3} - \frac{46}{375} a^{2} - \frac{4}{25} a + \frac{26}{75}$, $\frac{1}{1875} a^{16} + \frac{2}{1875} a^{15} - \frac{2}{625} a^{14} + \frac{172}{1875} a^{13} + \frac{149}{1875} a^{12} - \frac{58}{625} a^{11} - \frac{22}{375} a^{10} + \frac{206}{1875} a^{9} - \frac{446}{1875} a^{8} + \frac{149}{625} a^{7} + \frac{518}{1875} a^{6} - \frac{698}{1875} a^{5} + \frac{192}{625} a^{4} + \frac{43}{375} a^{3} + \frac{163}{1875} a^{2} + \frac{34}{125} a - \frac{46}{125}$, $\frac{1}{1875} a^{17} + \frac{3}{625} a^{14} - \frac{7}{125} a^{13} + \frac{6}{625} a^{12} - \frac{59}{625} a^{11} - \frac{3}{625} a^{10} - \frac{38}{1875} a^{9} - \frac{172}{625} a^{8} + \frac{33}{625} a^{7} + \frac{162}{625} a^{6} + \frac{154}{625} a^{5} + \frac{226}{625} a^{4} - \frac{54}{625} a^{3} + \frac{83}{625} a^{2} + \frac{38}{375} a - \frac{38}{125}$, $\frac{1}{1739571401218921430625} a^{18} - \frac{1}{193285711246546825625} a^{17} - \frac{149448516330949487}{1739571401218921430625} a^{16} + \frac{47823525225903844}{69582856048756857225} a^{15} - \frac{9097090915189449664}{1739571401218921430625} a^{14} + \frac{14252281373331071728}{579857133739640476875} a^{13} - \frac{169770798841168615817}{1739571401218921430625} a^{12} + \frac{169270799187781236587}{1739571401218921430625} a^{11} - \frac{126051275849230858652}{1739571401218921430625} a^{10} + \frac{245099819518979615659}{1739571401218921430625} a^{9} - \frac{2462585248097114361}{7731428449861873025} a^{8} + \frac{13593845240138976601}{56115206490932949375} a^{7} - \frac{1523981369874731462}{6235022943436994375} a^{6} - \frac{687347018883672141094}{1739571401218921430625} a^{5} - \frac{20767621816365620111}{56115206490932949375} a^{4} + \frac{47166130724630355049}{579857133739640476875} a^{3} - \frac{456880393405588799}{964820522029351875} a^{2} + \frac{27137507793823248718}{69582856048756857225} a - \frac{75115353440229233}{708583055486322375}$, $\frac{1}{47401581111814390063100625} a^{19} + \frac{2723}{9480316222362878012620125} a^{18} - \frac{12281059655414924659934}{47401581111814390063100625} a^{17} - \frac{11410948908078698869666}{47401581111814390063100625} a^{16} + \frac{1980172827902656619188}{3160105407454292670873375} a^{15} + \frac{139918575096007712626187}{47401581111814390063100625} a^{14} - \frac{1545936811443508392501452}{47401581111814390063100625} a^{13} + \frac{1033434499995846057772303}{15800527037271463354366875} a^{12} - \frac{186275170375601882507366}{3160105407454292670873375} a^{11} - \frac{521229753468467280904034}{9480316222362878012620125} a^{10} + \frac{17629663862167208858239456}{47401581111814390063100625} a^{9} - \frac{3905127050020405353312316}{9480316222362878012620125} a^{8} + \frac{92376674432131726168456}{1529083261671431937519375} a^{7} + \frac{583893253313998042219364}{47401581111814390063100625} a^{6} - \frac{621109341281538375302126}{3160105407454292670873375} a^{5} + \frac{15929771607765800243862352}{47401581111814390063100625} a^{4} + \frac{5466378362766664214783726}{15800527037271463354366875} a^{3} - \frac{13002438102230151138214151}{47401581111814390063100625} a^{2} - \frac{1832806410438421274122457}{9480316222362878012620125} a - \frac{577907138165800440449}{19308179678946798396375}$

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

Class group and class number

$C_{2}\times C_{22}\times C_{88}$, which has order $3872$ (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:  \( 6161971.68016 \) (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

\(\Q(\sqrt{-3255}) \), \(\Q(\sqrt{-651}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-651})\), 5.1.10595025.1 x5, 10.0.365388575713284375.1, 10.0.73077715142656875.1 x5, 10.2.561272773753125.1 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 R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$