↑

Properties

Label 20.0.45259984731...1081.3
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 11^{18}\cdot 13^{10}$
Root discriminant $54.05$
Ramified primes $3, 11, 13$
Class number $5500$ (GRH)
Class group $[5, 1100]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9340321, 1641938, -383733, 4167463, 2003270, 1386459, 2757163, -32737, 1791648, -144089, 637946, -48422, 135642, -8722, 17602, -888, 1358, -47, 57, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 57*x^18 - 47*x^17 + 1358*x^16 - 888*x^15 + 17602*x^14 - 8722*x^13 + 135642*x^12 - 48422*x^11 + 637946*x^10 - 144089*x^9 + 1791648*x^8 - 32737*x^7 + 2757163*x^6 + 1386459*x^5 + 2003270*x^4 + 4167463*x^3 - 383733*x^2 + 1641938*x + 9340321)
 
gp: K = bnfinit(x^20 - x^19 + 57*x^18 - 47*x^17 + 1358*x^16 - 888*x^15 + 17602*x^14 - 8722*x^13 + 135642*x^12 - 48422*x^11 + 637946*x^10 - 144089*x^9 + 1791648*x^8 - 32737*x^7 + 2757163*x^6 + 1386459*x^5 + 2003270*x^4 + 4167463*x^3 - 383733*x^2 + 1641938*x + 9340321, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 57 x^{18} - 47 x^{17} + 1358 x^{16} - 888 x^{15} + 17602 x^{14} - 8722 x^{13} + 135642 x^{12} - 48422 x^{11} + 637946 x^{10} - 144089 x^{9} + 1791648 x^{8} - 32737 x^{7} + 2757163 x^{6} + 1386459 x^{5} + 2003270 x^{4} + 4167463 x^{3} - 383733 x^{2} + 1641938 x + 9340321 \)

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:  \(45259984731914299465343386928991081=3^{10}\cdot 11^{18}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.05$
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:  $3, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(429=3\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{429}(1,·)$, $\chi_{429}(131,·)$, $\chi_{429}(196,·)$, $\chi_{429}(389,·)$, $\chi_{429}(326,·)$, $\chi_{429}(142,·)$, $\chi_{429}(337,·)$, $\chi_{429}(259,·)$, $\chi_{429}(404,·)$, $\chi_{429}(155,·)$, $\chi_{429}(157,·)$, $\chi_{429}(350,·)$, $\chi_{429}(415,·)$, $\chi_{429}(376,·)$, $\chi_{429}(38,·)$, $\chi_{429}(235,·)$, $\chi_{429}(365,·)$, $\chi_{429}(311,·)$, $\chi_{429}(248,·)$, $\chi_{429}(313,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2683} a^{11} + \frac{33}{2683} a^{9} + \frac{396}{2683} a^{7} - \frac{604}{2683} a^{5} - \frac{911}{2683} a^{3} - \frac{10}{2683} a + \frac{794}{2683}$, $\frac{1}{2683} a^{12} + \frac{33}{2683} a^{10} + \frac{396}{2683} a^{8} - \frac{604}{2683} a^{6} - \frac{911}{2683} a^{4} - \frac{10}{2683} a^{2} + \frac{794}{2683} a$, $\frac{1}{2683} a^{13} - \frac{693}{2683} a^{9} - \frac{257}{2683} a^{7} + \frac{240}{2683} a^{5} + \frac{540}{2683} a^{3} + \frac{794}{2683} a^{2} + \frac{330}{2683} a + \frac{628}{2683}$, $\frac{1}{2683} a^{14} - \frac{693}{2683} a^{10} - \frac{257}{2683} a^{8} + \frac{240}{2683} a^{6} + \frac{540}{2683} a^{4} + \frac{794}{2683} a^{3} + \frac{330}{2683} a^{2} + \frac{628}{2683} a$, $\frac{1}{10732} a^{15} + \frac{1}{10732} a^{13} - \frac{1}{10732} a^{12} - \frac{1}{5366} a^{11} - \frac{679}{2683} a^{10} - \frac{1147}{5366} a^{9} + \frac{2485}{5366} a^{8} - \frac{47}{10732} a^{7} + \frac{3287}{10732} a^{6} - \frac{719}{10732} a^{5} + \frac{1705}{10732} a^{4} - \frac{809}{10732} a^{3} + \frac{4115}{10732} a^{2} + \frac{675}{10732} a - \frac{733}{10732}$, $\frac{1}{249650241628} a^{16} - \frac{10367465}{249650241628} a^{15} + \frac{39756321}{249650241628} a^{14} - \frac{22763859}{124825120814} a^{13} + \frac{35090623}{249650241628} a^{12} - \frac{1707115}{124825120814} a^{11} - \frac{3308152867}{124825120814} a^{10} + \frac{23210895881}{62412560407} a^{9} + \frac{96960010843}{249650241628} a^{8} - \frac{24703499647}{124825120814} a^{7} + \frac{10449501587}{124825120814} a^{6} + \frac{25940877600}{62412560407} a^{5} - \frac{11785974041}{124825120814} a^{4} - \frac{27369762122}{62412560407} a^{3} - \frac{30994023770}{62412560407} a^{2} + \frac{8808699187}{62412560407} a + \frac{44379560293}{249650241628}$, $\frac{1}{249650241628} a^{17} + \frac{718023}{124825120814} a^{15} + \frac{44236383}{249650241628} a^{14} - \frac{31309489}{249650241628} a^{13} - \frac{34956473}{249650241628} a^{12} + \frac{1093436}{62412560407} a^{11} + \frac{22069374675}{124825120814} a^{10} + \frac{59769600323}{249650241628} a^{9} - \frac{22268423719}{249650241628} a^{8} - \frac{44570091553}{124825120814} a^{7} + \frac{27254451083}{62412560407} a^{6} + \frac{21193003321}{62412560407} a^{5} + \frac{23525750783}{62412560407} a^{4} - \frac{43818131485}{124825120814} a^{3} + \frac{60636954149}{124825120814} a^{2} - \frac{61840453381}{249650241628} a - \frac{55142456125}{249650241628}$, $\frac{1}{249650241628} a^{18} + \frac{2124109}{249650241628} a^{15} - \frac{6715051}{249650241628} a^{14} - \frac{27971685}{249650241628} a^{13} + \frac{6311939}{124825120814} a^{12} - \frac{816521}{124825120814} a^{11} + \frac{60547620759}{249650241628} a^{10} - \frac{61086105415}{249650241628} a^{9} - \frac{9369090581}{62412560407} a^{8} - \frac{6829781697}{62412560407} a^{7} + \frac{26332497404}{62412560407} a^{6} - \frac{16783450612}{62412560407} a^{5} + \frac{37696676473}{124825120814} a^{4} - \frac{14320773}{124825120814} a^{3} + \frac{99821219803}{249650241628} a^{2} + \frac{55678889187}{249650241628} a - \frac{51164334009}{124825120814}$, $\frac{1}{249650241628} a^{19} + \frac{5597831}{124825120814} a^{15} + \frac{17823937}{124825120814} a^{14} + \frac{4182356}{62412560407} a^{13} - \frac{33894645}{249650241628} a^{12} + \frac{29983389}{249650241628} a^{11} - \frac{110811595837}{249650241628} a^{10} - \frac{17114387483}{62412560407} a^{9} + \frac{29700469261}{249650241628} a^{8} - \frac{1414154601}{124825120814} a^{7} - \frac{60915834745}{124825120814} a^{6} + \frac{57704424515}{124825120814} a^{5} + \frac{30917910727}{62412560407} a^{4} + \frac{47764946539}{249650241628} a^{3} + \frac{117964472391}{249650241628} a^{2} + \frac{3456400283}{124825120814} a - \frac{25290200785}{249650241628}$

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

Class group and class number

$C_{5}\times C_{1100}$, which has order $5500$ (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:  \( 125582.779517 \) (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

\(\Q(\sqrt{-39}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-143}) \), \(\Q(\sqrt{33}, \sqrt{-39})\), \(\Q(\zeta_{11})^+\), 10.0.19340358336761319.3, \(\Q(\zeta_{33})^+\), 10.0.875489472034463.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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}$
$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}$
13Data not computed