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Properties

Label 20.0.45386099695...0784.6
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 31^{18}$
Root discriminant $152.35$
Ramified primes $2, 3, 31$
Class number $1093224$ (GRH)
Class group $[2, 2, 273306]$ (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![1093691041, 0, 1004707748, 0, 322730031, 0, 18355260, 0, -8987849, 0, -1497772, 0, -5464, 0, 16260, 0, 1901, 0, 68, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 68*x^18 + 1901*x^16 + 16260*x^14 - 5464*x^12 - 1497772*x^10 - 8987849*x^8 + 18355260*x^6 + 322730031*x^4 + 1004707748*x^2 + 1093691041)
 
gp: K = bnfinit(x^20 + 68*x^18 + 1901*x^16 + 16260*x^14 - 5464*x^12 - 1497772*x^10 - 8987849*x^8 + 18355260*x^6 + 322730031*x^4 + 1004707748*x^2 + 1093691041, 1)
 

Normalized defining polynomial

\( x^{20} + 68 x^{18} + 1901 x^{16} + 16260 x^{14} - 5464 x^{12} - 1497772 x^{10} - 8987849 x^{8} + 18355260 x^{6} + 322730031 x^{4} + 1004707748 x^{2} + 1093691041 \)

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:  \(45386099695641231028031241010795100749430784=2^{40}\cdot 3^{10}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $152.35$
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:  $2, 3, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(744=2^{3}\cdot 3\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{744}(1,·)$, $\chi_{744}(395,·)$, $\chi_{744}(97,·)$, $\chi_{744}(587,·)$, $\chi_{744}(529,·)$, $\chi_{744}(275,·)$, $\chi_{744}(469,·)$, $\chi_{744}(23,·)$, $\chi_{744}(721,·)$, $\chi_{744}(349,·)$, $\chi_{744}(481,·)$, $\chi_{744}(743,·)$, $\chi_{744}(263,·)$, $\chi_{744}(647,·)$, $\chi_{744}(109,·)$, $\chi_{744}(157,·)$, $\chi_{744}(371,·)$, $\chi_{744}(373,·)$, $\chi_{744}(635,·)$, $\chi_{744}(215,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{257490917289984707318478225711163803611045} a^{18} - \frac{106070565782162678431564357429668179580741}{257490917289984707318478225711163803611045} a^{16} - \frac{6859641892539751011583001075136914716936}{51498183457996941463695645142232760722209} a^{14} + \frac{17406985277632054714254372806570493234608}{51498183457996941463695645142232760722209} a^{12} + \frac{60934569619990468366275525578860093922946}{257490917289984707318478225711163803611045} a^{10} + \frac{51892750652113618956854391875105977073964}{257490917289984707318478225711163803611045} a^{8} - \frac{14304793053448711761656046370785790764407}{51498183457996941463695645142232760722209} a^{6} + \frac{8039332614883640202117146746413216412530}{51498183457996941463695645142232760722209} a^{4} - \frac{28204338199371408191130291694010931974799}{257490917289984707318478225711163803611045} a^{2} + \frac{54793624902464721527919562957649857264324}{257490917289984707318478225711163803611045}$, $\frac{1}{8515482125697084255729393402493898149220869195} a^{19} - \frac{407714192635827954363582595658201969295864976}{8515482125697084255729393402493898149220869195} a^{17} + \frac{266496239753241632323613380609979399822714639}{1703096425139416851145878680498779629844173839} a^{15} - \frac{831884248595204960349825197254821446213329578}{1703096425139416851145878680498779629844173839} a^{13} - \frac{646241267828241624901014071009442286969800004}{8515482125697084255729393402493898149220869195} a^{11} + \frac{51292585291359070375334021308396702895671919}{8515482125697084255729393402493898149220869195} a^{9} - \frac{764916823694682020272033073343953980797734684}{1703096425139416851145878680498779629844173839} a^{7} + \frac{247096323564084208783013822539179199161571312}{1703096425139416851145878680498779629844173839} a^{5} + \frac{2940260580196136001461511729104085462502548056}{8515482125697084255729393402493898149220869195} a^{3} + \frac{889428421944509643799551711169317427529813754}{8515482125697084255729393402493898149220869195} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{273306}$, which has order $1093224$ (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:  \( 24785765.76033278 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-93}) \), \(\Q(\sqrt{-186}) \), \(\Q(\sqrt{2}, \sqrt{-93})\), 5.5.923521.1, 10.10.27947533514866688.1, 10.0.6579024061484086272.1, 10.0.210528769967490760704.1

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
3Data not computed
$31$31.10.9.1$x^{10} - 31$$10$$1$$9$$C_{10}$$[\ ]_{10}$
31.10.9.1$x^{10} - 31$$10$$1$$9$$C_{10}$$[\ ]_{10}$