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Properties

Label 20.0.94964489254...000.16
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $177.37$
Ramified primes $2, 3, 5, 7, 11$
Class number $127315232$ (GRH)
Class group $[2, 44, 1446764]$ (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![2174170893049, 0, 681416044115, 0, 173004197745, 0, 31277505147, 0, 3828194857, 0, 313909091, 0, 17167649, 0, 616203, 0, 13913, 0, 179, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 179*x^18 + 13913*x^16 + 616203*x^14 + 17167649*x^12 + 313909091*x^10 + 3828194857*x^8 + 31277505147*x^6 + 173004197745*x^4 + 681416044115*x^2 + 2174170893049)
 
gp: K = bnfinit(x^20 + 179*x^18 + 13913*x^16 + 616203*x^14 + 17167649*x^12 + 313909091*x^10 + 3828194857*x^8 + 31277505147*x^6 + 173004197745*x^4 + 681416044115*x^2 + 2174170893049, 1)
 

Normalized defining polynomial

\( x^{20} + 179 x^{18} + 13913 x^{16} + 616203 x^{14} + 17167649 x^{12} + 313909091 x^{10} + 3828194857 x^{8} + 31277505147 x^{6} + 173004197745 x^{4} + 681416044115 x^{2} + 2174170893049 \)

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:  \(949644892545940254829738055123773440000000000=2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $177.37$
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:  $2, 3, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4620=2^{2}\cdot 3\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{4620}(1,·)$, $\chi_{4620}(139,·)$, $\chi_{4620}(2309,·)$, $\chi_{4620}(3079,·)$, $\chi_{4620}(841,·)$, $\chi_{4620}(3851,·)$, $\chi_{4620}(911,·)$, $\chi_{4620}(1681,·)$, $\chi_{4620}(3989,·)$, $\chi_{4620}(1469,·)$, $\chi_{4620}(1889,·)$, $\chi_{4620}(2659,·)$, $\chi_{4620}(421,·)$, $\chi_{4620}(71,·)$, $\chi_{4620}(4271,·)$, $\chi_{4620}(629,·)$, $\chi_{4620}(1399,·)$, $\chi_{4620}(2171,·)$, $\chi_{4620}(2941,·)$, $\chi_{4620}(2239,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{1474507} a^{11} + \frac{88}{1474507} a^{9} + \frac{2816}{1474507} a^{7} + \frac{39424}{1474507} a^{5} + \frac{225280}{1474507} a^{3} + \frac{360448}{1474507} a$, $\frac{1}{638206441289} a^{12} + \frac{189751356904}{638206441289} a^{10} - \frac{136846042840}{638206441289} a^{8} - \frac{2450591210}{638206441289} a^{6} + \frac{245510064301}{638206441289} a^{4} - \frac{294373166046}{638206441289} a^{2} + \frac{62673}{432827}$, $\frac{1}{638206441289} a^{13} + \frac{104}{638206441289} a^{11} - \frac{241597967726}{638206441289} a^{9} - \frac{163479981117}{638206441289} a^{7} - \frac{94282070530}{638206441289} a^{5} + \frac{225610908463}{638206441289} a^{3} - \frac{96744009637}{638206441289} a$, $\frac{1}{638206441289} a^{14} - \frac{191339405783}{638206441289} a^{10} + \frac{27966765885}{638206441289} a^{8} + \frac{160579415310}{638206441289} a^{6} + \frac{220821872719}{638206441289} a^{4} - \frac{115843922725}{638206441289} a^{2} - \frac{25587}{432827}$, $\frac{1}{638206441289} a^{15} - \frac{6720}{638206441289} a^{11} + \frac{272466409915}{638206441289} a^{9} - \frac{13570248369}{27748106143} a^{7} - \frac{14845503549}{638206441289} a^{5} - \frac{277274110434}{638206441289} a^{3} + \frac{86907353830}{638206441289} a$, $\frac{1}{638206441289} a^{16} + \frac{265115109373}{638206441289} a^{10} - \frac{262041699838}{638206441289} a^{8} + \frac{110549038765}{638206441289} a^{6} - \frac{213292739779}{638206441289} a^{4} - \frac{299006920679}{638206441289} a^{2} + \frac{21889}{432827}$, $\frac{1}{638206441289} a^{17} - \frac{84667}{638206441289} a^{11} + \frac{21459552335}{638206441289} a^{9} + \frac{247698930255}{638206441289} a^{7} - \frac{207813582786}{638206441289} a^{5} - \frac{176525103392}{638206441289} a^{3} - \frac{282318761649}{638206441289} a$, $\frac{1}{638206441289} a^{18} + \frac{128847975306}{638206441289} a^{10} - \frac{96475043519}{638206441289} a^{8} - \frac{274926140931}{638206441289} a^{6} + \frac{40296286645}{638206441289} a^{4} - \frac{99016719014}{638206441289} a^{2} - \frac{124129}{432827}$, $\frac{1}{638206441289} a^{19} + \frac{138503}{638206441289} a^{11} + \frac{52631261019}{638206441289} a^{9} + \frac{29030515262}{638206441289} a^{7} - \frac{171755615676}{638206441289} a^{5} - \frac{34328992556}{638206441289} a^{3} - \frac{9007391536}{27748106143} a$

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

Class group and class number

$C_{2}\times C_{44}\times C_{1446764}$, which has order $127315232$ (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:  \( 1746210.0427691017 \) (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

\(\Q(\sqrt{-385}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1155}) \), \(\Q(\sqrt{3}, \sqrt{-385})\), \(\Q(\zeta_{11})^+\), 10.0.126816085896438400000.1, 10.10.53339349076992.1, 10.0.30094051633627471875.1

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 R R R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
3Data not computed
5Data not computed
7Data 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}$