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Properties

Label 20.0.10019151533...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{15}\cdot 11^{18}$
Root discriminant $50.12$
Ramified primes $3, 5, 11$
Class number $3368$ (GRH)
Class group $[2, 2, 842]$ (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![1197901, 173262, 1371042, -411001, 958226, -492583, 456689, -254178, 181094, -88253, 64285, -23144, 18734, -4410, 3874, -536, 500, -36, 35, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 35*x^18 - 36*x^17 + 500*x^16 - 536*x^15 + 3874*x^14 - 4410*x^13 + 18734*x^12 - 23144*x^11 + 64285*x^10 - 88253*x^9 + 181094*x^8 - 254178*x^7 + 456689*x^6 - 492583*x^5 + 958226*x^4 - 411001*x^3 + 1371042*x^2 + 173262*x + 1197901)
 
gp: K = bnfinit(x^20 - x^19 + 35*x^18 - 36*x^17 + 500*x^16 - 536*x^15 + 3874*x^14 - 4410*x^13 + 18734*x^12 - 23144*x^11 + 64285*x^10 - 88253*x^9 + 181094*x^8 - 254178*x^7 + 456689*x^6 - 492583*x^5 + 958226*x^4 - 411001*x^3 + 1371042*x^2 + 173262*x + 1197901, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 35 x^{18} - 36 x^{17} + 500 x^{16} - 536 x^{15} + 3874 x^{14} - 4410 x^{13} + 18734 x^{12} - 23144 x^{11} + 64285 x^{10} - 88253 x^{9} + 181094 x^{8} - 254178 x^{7} + 456689 x^{6} - 492583 x^{5} + 958226 x^{4} - 411001 x^{3} + 1371042 x^{2} + 173262 x + 1197901 \)

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:  \(10019151533337487082567413330078125=3^{10}\cdot 5^{15}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.12$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(165=3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{165}(128,·)$, $\chi_{165}(1,·)$, $\chi_{165}(2,·)$, $\chi_{165}(4,·)$, $\chi_{165}(136,·)$, $\chi_{165}(98,·)$, $\chi_{165}(16,·)$, $\chi_{165}(17,·)$, $\chi_{165}(83,·)$, $\chi_{165}(68,·)$, $\chi_{165}(91,·)$, $\chi_{165}(31,·)$, $\chi_{165}(32,·)$, $\chi_{165}(34,·)$, $\chi_{165}(64,·)$, $\chi_{165}(107,·)$, $\chi_{165}(49,·)$, $\chi_{165}(8,·)$, $\chi_{165}(124,·)$, $\chi_{165}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{89} a^{15} + \frac{42}{89} a^{14} - \frac{14}{89} a^{13} - \frac{44}{89} a^{12} + \frac{5}{89} a^{11} - \frac{19}{89} a^{10} + \frac{14}{89} a^{9} - \frac{34}{89} a^{8} - \frac{41}{89} a^{7} - \frac{36}{89} a^{6} - \frac{31}{89} a^{4} + \frac{7}{89} a^{3} - \frac{32}{89} a^{2} - \frac{43}{89} a - \frac{6}{89}$, $\frac{1}{184141} a^{16} - \frac{851}{184141} a^{15} + \frac{61270}{184141} a^{14} + \frac{70308}{184141} a^{13} + \frac{5388}{184141} a^{12} + \frac{53188}{184141} a^{11} + \frac{8793}{184141} a^{10} + \frac{27959}{184141} a^{9} + \frac{49990}{184141} a^{8} + \frac{33106}{184141} a^{7} - \frac{47507}{184141} a^{6} + \frac{41888}{184141} a^{5} + \frac{10068}{184141} a^{4} - \frac{5304}{184141} a^{3} - \frac{56195}{184141} a^{2} + \frac{27713}{184141} a + \frac{75935}{184141}$, $\frac{1}{184141} a^{17} - \frac{851}{184141} a^{15} - \frac{82897}{184141} a^{14} - \frac{791}{2069} a^{13} - \frac{2391}{184141} a^{12} - \frac{31043}{184141} a^{11} + \frac{87230}{184141} a^{10} - \frac{33161}{184141} a^{9} - \frac{7493}{184141} a^{8} + \frac{59714}{184141} a^{7} + \frac{43760}{184141} a^{6} - \frac{66598}{184141} a^{5} + \frac{7249}{184141} a^{4} + \frac{64661}{184141} a^{3} + \frac{72083}{184141} a^{2} - \frac{22076}{184141} a + \frac{65816}{184141}$, $\frac{1}{184141} a^{18} - \frac{188}{184141} a^{15} - \frac{33256}{184141} a^{14} - \frac{80247}{184141} a^{13} - \frac{14207}{184141} a^{12} + \frac{34980}{184141} a^{11} + \frac{36455}{184141} a^{10} - \frac{88575}{184141} a^{9} + \frac{66702}{184141} a^{8} - \frac{78678}{184141} a^{7} + \frac{61483}{184141} a^{6} - \frac{69417}{184141} a^{5} + \frac{6868}{184141} a^{4} - \frac{82238}{184141} a^{3} - \frac{8741}{184141} a^{2} + \frac{909}{184141} a - \frac{66600}{184141}$, $\frac{1}{4400480492326040721932357141099143591} a^{19} - \frac{4947378899718961503897270439861}{4400480492326040721932357141099143591} a^{18} + \frac{5332784664474912245459306460029}{4400480492326040721932357141099143591} a^{17} + \frac{5546832287831780196686860333186}{4400480492326040721932357141099143591} a^{16} - \frac{19699994265825131124003396122327418}{4400480492326040721932357141099143591} a^{15} + \frac{1544260033225722969716088028645530905}{4400480492326040721932357141099143591} a^{14} - \frac{729913725477769105840537823892929217}{4400480492326040721932357141099143591} a^{13} - \frac{1711694317370337956544684059136280988}{4400480492326040721932357141099143591} a^{12} + \frac{471606585701340388563666222205211519}{4400480492326040721932357141099143591} a^{11} + \frac{116690532758686955320686899821500698}{4400480492326040721932357141099143591} a^{10} - \frac{2056228016115939045074101577635670547}{4400480492326040721932357141099143591} a^{9} + \frac{1442797088526163948888296296697034050}{4400480492326040721932357141099143591} a^{8} + \frac{1263976166726037453371739763806673610}{4400480492326040721932357141099143591} a^{7} - \frac{457172815360481636849748629071708133}{4400480492326040721932357141099143591} a^{6} - \frac{1748255277368910891864649476570277171}{4400480492326040721932357141099143591} a^{5} + \frac{314934474600265553705073950450810274}{4400480492326040721932357141099143591} a^{4} - \frac{1190087888218587861368733548045447439}{4400480492326040721932357141099143591} a^{3} - \frac{1328965794604476922546575123056478954}{4400480492326040721932357141099143591} a^{2} - \frac{1709278034191080005847341203869061575}{4400480492326040721932357141099143591} a - \frac{1125243369228987574069809791759}{3673492627793148784358938794691}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{842}$, which has order $3368$ (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:  \( 140644.599182 \) (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

\(\Q(\sqrt{5}) \), 4.0.136125.2, \(\Q(\zeta_{11})^+\), 10.10.669871503125.1

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 $20$ R R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
3Data not computed
5Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$