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Properties

Label 20.0.58299958569...0000.5
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{10}\cdot 11^{18}$
Root discriminant $54.74$
Ramified primes $2, 5, 11$
Class number $1464$ (GRH)
Class group $[1464]$ (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![1916641, 1333444, 2512812, -720158, 2282848, -598132, 579913, -296394, 150453, -43646, 16774, -10496, 4758, -980, 378, -224, 106, -12, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 5*x^18 - 12*x^17 + 106*x^16 - 224*x^15 + 378*x^14 - 980*x^13 + 4758*x^12 - 10496*x^11 + 16774*x^10 - 43646*x^9 + 150453*x^8 - 296394*x^7 + 579913*x^6 - 598132*x^5 + 2282848*x^4 - 720158*x^3 + 2512812*x^2 + 1333444*x + 1916641)
 
gp: K = bnfinit(x^20 - 2*x^19 + 5*x^18 - 12*x^17 + 106*x^16 - 224*x^15 + 378*x^14 - 980*x^13 + 4758*x^12 - 10496*x^11 + 16774*x^10 - 43646*x^9 + 150453*x^8 - 296394*x^7 + 579913*x^6 - 598132*x^5 + 2282848*x^4 - 720158*x^3 + 2512812*x^2 + 1333444*x + 1916641, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 5 x^{18} - 12 x^{17} + 106 x^{16} - 224 x^{15} + 378 x^{14} - 980 x^{13} + 4758 x^{12} - 10496 x^{11} + 16774 x^{10} - 43646 x^{9} + 150453 x^{8} - 296394 x^{7} + 579913 x^{6} - 598132 x^{5} + 2282848 x^{4} - 720158 x^{3} + 2512812 x^{2} + 1333444 x + 1916641 \)

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:  \(58299958569124301174210560000000000=2^{30}\cdot 5^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.74$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(440=2^{3}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{440}(1,·)$, $\chi_{440}(129,·)$, $\chi_{440}(201,·)$, $\chi_{440}(141,·)$, $\chi_{440}(109,·)$, $\chi_{440}(81,·)$, $\chi_{440}(149,·)$, $\chi_{440}(409,·)$, $\chi_{440}(221,·)$, $\chi_{440}(421,·)$, $\chi_{440}(401,·)$, $\chi_{440}(361,·)$, $\chi_{440}(29,·)$, $\chi_{440}(301,·)$, $\chi_{440}(349,·)$, $\chi_{440}(369,·)$, $\chi_{440}(181,·)$, $\chi_{440}(329,·)$, $\chi_{440}(249,·)$, $\chi_{440}(189,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{89} a^{12} - \frac{21}{89} a^{11} - \frac{1}{89} a^{10} - \frac{21}{89} a^{9} + \frac{42}{89} a^{8} - \frac{44}{89} a^{7} - \frac{42}{89} a^{6} + \frac{3}{89} a^{5} + \frac{42}{89} a^{4} - \frac{19}{89} a^{3} - \frac{32}{89} a^{2} + \frac{12}{89} a + \frac{44}{89}$, $\frac{1}{89} a^{13} + \frac{3}{89} a^{11} - \frac{42}{89} a^{10} - \frac{43}{89} a^{9} + \frac{37}{89} a^{8} + \frac{13}{89} a^{7} + \frac{11}{89} a^{6} + \frac{16}{89} a^{5} - \frac{27}{89} a^{4} + \frac{14}{89} a^{3} - \frac{37}{89} a^{2} + \frac{29}{89} a + \frac{34}{89}$, $\frac{1}{89} a^{14} + \frac{21}{89} a^{11} - \frac{40}{89} a^{10} + \frac{11}{89} a^{9} - \frac{24}{89} a^{8} - \frac{35}{89} a^{7} - \frac{36}{89} a^{6} - \frac{36}{89} a^{5} - \frac{23}{89} a^{4} + \frac{20}{89} a^{3} + \frac{36}{89} a^{2} - \frac{2}{89} a - \frac{43}{89}$, $\frac{1}{89} a^{15} - \frac{44}{89} a^{11} + \frac{32}{89} a^{10} - \frac{28}{89} a^{9} - \frac{27}{89} a^{8} - \frac{2}{89} a^{7} - \frac{44}{89} a^{6} + \frac{3}{89} a^{5} + \frac{28}{89} a^{4} - \frac{10}{89} a^{3} - \frac{42}{89} a^{2} - \frac{28}{89} a - \frac{34}{89}$, $\frac{1}{89} a^{16} - \frac{2}{89} a^{11} + \frac{17}{89} a^{10} + \frac{28}{89} a^{9} - \frac{23}{89} a^{8} - \frac{22}{89} a^{7} + \frac{24}{89} a^{6} - \frac{18}{89} a^{5} - \frac{31}{89} a^{4} + \frac{12}{89} a^{3} - \frac{12}{89} a^{2} - \frac{40}{89} a - \frac{22}{89}$, $\frac{1}{89} a^{17} - \frac{25}{89} a^{11} + \frac{26}{89} a^{10} + \frac{24}{89} a^{9} - \frac{27}{89} a^{8} + \frac{25}{89} a^{7} - \frac{13}{89} a^{6} - \frac{25}{89} a^{5} + \frac{7}{89} a^{4} + \frac{39}{89} a^{3} - \frac{15}{89} a^{2} + \frac{2}{89} a - \frac{1}{89}$, $\frac{1}{17711} a^{18} - \frac{2}{17711} a^{17} + \frac{86}{17711} a^{16} + \frac{25}{17711} a^{15} - \frac{92}{17711} a^{14} + \frac{10}{17711} a^{13} + \frac{90}{17711} a^{12} + \frac{7392}{17711} a^{11} - \frac{3076}{17711} a^{10} - \frac{1779}{17711} a^{9} - \frac{3532}{17711} a^{8} + \frac{3494}{17711} a^{7} + \frac{5253}{17711} a^{6} - \frac{7211}{17711} a^{5} + \frac{6426}{17711} a^{4} - \frac{6133}{17711} a^{3} + \frac{6608}{17711} a^{2} + \frac{3138}{17711} a + \frac{2166}{17711}$, $\frac{1}{4626918497284982027174856006336931541614317690433} a^{19} + \frac{68325175361520303280349242236646068408576237}{4626918497284982027174856006336931541614317690433} a^{18} - \frac{3425717053818777753003279914913244645543052271}{4626918497284982027174856006336931541614317690433} a^{17} + \frac{15128558588796786577326142281542547370297734199}{4626918497284982027174856006336931541614317690433} a^{16} - \frac{570979273705136970262783095778112011919122182}{4626918497284982027174856006336931541614317690433} a^{15} + \frac{5671918911143885867598243583292737301141283887}{4626918497284982027174856006336931541614317690433} a^{14} - \frac{25074187715905465077195743974468362103033199808}{4626918497284982027174856006336931541614317690433} a^{13} - \frac{3230321004700928160814379417396336711506468082}{4626918497284982027174856006336931541614317690433} a^{12} + \frac{930055436459408824838706317281065598455293358341}{4626918497284982027174856006336931541614317690433} a^{11} + \frac{128964816764558304122618824940673569346845503564}{4626918497284982027174856006336931541614317690433} a^{10} - \frac{1692344154995720052285478437511982085038639799676}{4626918497284982027174856006336931541614317690433} a^{9} + \frac{1620548840695108092253989908438330725333506975689}{4626918497284982027174856006336931541614317690433} a^{8} - \frac{1574477731288160853803916570427567105035337608677}{4626918497284982027174856006336931541614317690433} a^{7} + \frac{912915206025447167948524571516636832520503038585}{4626918497284982027174856006336931541614317690433} a^{6} + \frac{1572333822065202617756520408651332953037891625589}{4626918497284982027174856006336931541614317690433} a^{5} + \frac{1575365677919175372388275647449544028355271255113}{4626918497284982027174856006336931541614317690433} a^{4} - \frac{1169573375012268714115527952013889703456793404824}{4626918497284982027174856006336931541614317690433} a^{3} + \frac{844501789107098496399045884504168483703227690244}{4626918497284982027174856006336931541614317690433} a^{2} - \frac{361808877283382478741059256466607512160953614007}{4626918497284982027174856006336931541614317690433} a + \frac{1372384834645498468304454868659065316834186404287}{4626918497284982027174856006336931541614317690433}$

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

Class group and class number

$C_{1464}$, which has order $1464$ (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:  \( 530208.250733 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{2}, \sqrt{-55})\), \(\Q(\zeta_{11})^+\), 10.10.7024111812608.1, 10.0.241453843558400000.1, 10.0.7368586534375.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
5Data not computed
11Data not computed