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Properties

Label 20.0.86825139158...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{15}\cdot 11^{16}$
Root discriminant $78.87$
Ramified primes $2, 3, 5, 11$
Class number $318100$ (GRH)
Class group $[10, 31810]$ (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![2065660201, -786154512, 1764047967, -597365054, 733497061, -224863132, 196375269, -53466672, 36960084, -9006872, 5168310, -1102236, 534824, -96080, 39824, -5724, 2035, -214, 65, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 65*x^18 - 214*x^17 + 2035*x^16 - 5724*x^15 + 39824*x^14 - 96080*x^13 + 534824*x^12 - 1102236*x^11 + 5168310*x^10 - 9006872*x^9 + 36960084*x^8 - 53466672*x^7 + 196375269*x^6 - 224863132*x^5 + 733497061*x^4 - 597365054*x^3 + 1764047967*x^2 - 786154512*x + 2065660201)
 
gp: K = bnfinit(x^20 - 4*x^19 + 65*x^18 - 214*x^17 + 2035*x^16 - 5724*x^15 + 39824*x^14 - 96080*x^13 + 534824*x^12 - 1102236*x^11 + 5168310*x^10 - 9006872*x^9 + 36960084*x^8 - 53466672*x^7 + 196375269*x^6 - 224863132*x^5 + 733497061*x^4 - 597365054*x^3 + 1764047967*x^2 - 786154512*x + 2065660201, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 65 x^{18} - 214 x^{17} + 2035 x^{16} - 5724 x^{15} + 39824 x^{14} - 96080 x^{13} + 534824 x^{12} - 1102236 x^{11} + 5168310 x^{10} - 9006872 x^{9} + 36960084 x^{8} - 53466672 x^{7} + 196375269 x^{6} - 224863132 x^{5} + 733497061 x^{4} - 597365054 x^{3} + 1764047967 x^{2} - 786154512 x + 2065660201 \)

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:  \(86825139158850321116448000000000000000=2^{20}\cdot 3^{10}\cdot 5^{15}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.87$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(660=2^{2}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{660}(1,·)$, $\chi_{660}(323,·)$, $\chi_{660}(647,·)$, $\chi_{660}(587,·)$, $\chi_{660}(529,·)$, $\chi_{660}(467,·)$, $\chi_{660}(203,·)$, $\chi_{660}(23,·)$, $\chi_{660}(229,·)$, $\chi_{660}(289,·)$, $\chi_{660}(421,·)$, $\chi_{660}(169,·)$, $\chi_{660}(301,·)$, $\chi_{660}(47,·)$, $\chi_{660}(287,·)$, $\chi_{660}(49,·)$, $\chi_{660}(181,·)$, $\chi_{660}(361,·)$, $\chi_{660}(443,·)$, $\chi_{660}(383,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{27347951517410290260783} a^{18} - \frac{1827910011980938465684}{27347951517410290260783} a^{17} + \frac{493800810277204215802}{9115983839136763420261} a^{16} + \frac{223071515770156019477}{27347951517410290260783} a^{15} + \frac{1291927620507676731599}{9115983839136763420261} a^{14} + \frac{2714060910309887463764}{27347951517410290260783} a^{13} + \frac{1485556879593325920793}{27347951517410290260783} a^{12} + \frac{762479911029048605051}{27347951517410290260783} a^{11} + \frac{3350665101475949812465}{27347951517410290260783} a^{10} - \frac{1724206235215389963128}{9115983839136763420261} a^{9} - \frac{7365960566265552459073}{27347951517410290260783} a^{8} + \frac{3677654216964824422221}{9115983839136763420261} a^{7} - \frac{10206254912513863664860}{27347951517410290260783} a^{6} - \frac{11043451838754507157523}{27347951517410290260783} a^{5} - \frac{8408760488448597868387}{27347951517410290260783} a^{4} - \frac{11468290887127534440917}{27347951517410290260783} a^{3} - \frac{10026698900722160686135}{27347951517410290260783} a^{2} + \frac{9328724508980657338415}{27347951517410290260783} a - \frac{11757387029511197785337}{27347951517410290260783}$, $\frac{1}{23165855440749749943492156412021652491938659083368843} a^{19} - \frac{1115561594114307246483303575}{7721951813583249981164052137340550830646219694456281} a^{18} + \frac{3056535228854305712321741613591603752511732713028487}{23165855440749749943492156412021652491938659083368843} a^{17} + \frac{2304207189702925833432943849300468206374344795913795}{23165855440749749943492156412021652491938659083368843} a^{16} + \frac{1486251729372586357760876632659618664662594959134786}{23165855440749749943492156412021652491938659083368843} a^{15} + \frac{733071844824178482837615134785850279773545826189555}{23165855440749749943492156412021652491938659083368843} a^{14} - \frac{329498528065891686958112199644496743597897094007400}{7721951813583249981164052137340550830646219694456281} a^{13} + \frac{1784515731984570048291125914416678620415563856626327}{23165855440749749943492156412021652491938659083368843} a^{12} - \frac{1013881306278993555007049083751131865425353248258954}{7721951813583249981164052137340550830646219694456281} a^{11} + \frac{491269250436375203998990611762205682087038744084687}{23165855440749749943492156412021652491938659083368843} a^{10} - \frac{591050411036856820430296440516402611520824990356137}{7721951813583249981164052137340550830646219694456281} a^{9} + \frac{2498591921793791377727770601557347118636837736391761}{23165855440749749943492156412021652491938659083368843} a^{8} - \frac{6360442005725515908267699977796102746513711765288791}{23165855440749749943492156412021652491938659083368843} a^{7} + \frac{2478144828598876598698258992827990007497948250135533}{23165855440749749943492156412021652491938659083368843} a^{6} - \frac{3494836633885577040151975343217691868982016563659728}{7721951813583249981164052137340550830646219694456281} a^{5} - \frac{280493297074722508160230108841882241356016499674507}{7721951813583249981164052137340550830646219694456281} a^{4} + \frac{1203706812023482192825305595242434996129512917815970}{7721951813583249981164052137340550830646219694456281} a^{3} - \frac{1903618638079985966659505214544153267204004195526427}{23165855440749749943492156412021652491938659083368843} a^{2} - \frac{8818033542316041920005635157265137702670146163565022}{23165855440749749943492156412021652491938659083368843} a + \frac{2714000807395014425382086440684675343202509510976256}{23165855440749749943492156412021652491938659083368843}$

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

Class group and class number

$C_{10}\times C_{31810}$, which has order $318100$ (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

\(\Q(\sqrt{5}) \), 4.0.18000.1, \(\Q(\zeta_{11})^+\), 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 R R R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$