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Properties

Label 20.0.26496929674...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 11^{16}$
Root discriminant $26.37$
Ramified primes $3, 5, 11$
Class number $11$
Class group $[11]$
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![1, 3, 27, -40, 293, -175, 860, -114, 1892, 109, 1987, 223, 1496, 141, 547, -23, 131, -3, 14, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 14*x^18 - 3*x^17 + 131*x^16 - 23*x^15 + 547*x^14 + 141*x^13 + 1496*x^12 + 223*x^11 + 1987*x^10 + 109*x^9 + 1892*x^8 - 114*x^7 + 860*x^6 - 175*x^5 + 293*x^4 - 40*x^3 + 27*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 14*x^18 - 3*x^17 + 131*x^16 - 23*x^15 + 547*x^14 + 141*x^13 + 1496*x^12 + 223*x^11 + 1987*x^10 + 109*x^9 + 1892*x^8 - 114*x^7 + 860*x^6 - 175*x^5 + 293*x^4 - 40*x^3 + 27*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 14 x^{18} - 3 x^{17} + 131 x^{16} - 23 x^{15} + 547 x^{14} + 141 x^{13} + 1496 x^{12} + 223 x^{11} + 1987 x^{10} + 109 x^{9} + 1892 x^{8} - 114 x^{7} + 860 x^{6} - 175 x^{5} + 293 x^{4} - 40 x^{3} + 27 x^{2} + 3 x + 1 \)

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:  \(26496929674942114598525390625=3^{10}\cdot 5^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.37$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(165=3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{165}(64,·)$, $\chi_{165}(1,·)$, $\chi_{165}(4,·)$, $\chi_{165}(71,·)$, $\chi_{165}(136,·)$, $\chi_{165}(14,·)$, $\chi_{165}(16,·)$, $\chi_{165}(146,·)$, $\chi_{165}(86,·)$, $\chi_{165}(89,·)$, $\chi_{165}(26,·)$, $\chi_{165}(91,·)$, $\chi_{165}(31,·)$, $\chi_{165}(34,·)$, $\chi_{165}(104,·)$, $\chi_{165}(49,·)$, $\chi_{165}(119,·)$, $\chi_{165}(56,·)$, $\chi_{165}(59,·)$, $\chi_{165}(124,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{10} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{11} - \frac{1}{2} a^{2}$, $\frac{1}{8826446} a^{18} + \frac{2151931}{8826446} a^{17} + \frac{6263}{131738} a^{16} + \frac{1966329}{8826446} a^{15} + \frac{30927}{65869} a^{14} - \frac{1741872}{4413223} a^{13} - \frac{3815147}{8826446} a^{12} - \frac{3080109}{8826446} a^{11} + \frac{1836751}{8826446} a^{10} + \frac{3830505}{8826446} a^{9} + \frac{622378}{4413223} a^{8} - \frac{1730055}{4413223} a^{7} - \frac{668738}{4413223} a^{6} + \frac{281908}{4413223} a^{5} + \frac{46067}{4413223} a^{4} + \frac{2173069}{8826446} a^{3} - \frac{841273}{8826446} a^{2} - \frac{3461257}{8826446} a + \frac{3473023}{8826446}$, $\frac{1}{13946611855850460074} a^{19} - \frac{210472290587}{6973305927925230037} a^{18} - \frac{609036177295437235}{13946611855850460074} a^{17} + \frac{3864211555638067}{6973305927925230037} a^{16} + \frac{1494456998539333873}{6973305927925230037} a^{15} + \frac{454423146162384334}{6973305927925230037} a^{14} + \frac{6115638706187400203}{13946611855850460074} a^{13} + \frac{1191973854267582202}{6973305927925230037} a^{12} - \frac{29878592747036625}{70083476662565126} a^{11} - \frac{1327410353734991943}{6973305927925230037} a^{10} + \frac{1621296334334777832}{6973305927925230037} a^{9} + \frac{3091930014643155899}{6973305927925230037} a^{8} - \frac{371073757363280614}{6973305927925230037} a^{7} - \frac{3043662032713045206}{6973305927925230037} a^{6} - \frac{2195818172144761067}{6973305927925230037} a^{5} - \frac{18943485689313445}{42134779020696254} a^{4} - \frac{2606101588964526828}{6973305927925230037} a^{3} + \frac{4453702580964399311}{13946611855850460074} a^{2} + \frac{962909471896249902}{6973305927925230037} a - \frac{1620492261825237564}{6973305927925230037}$

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

Class group and class number

$C_{11}$, which has order $11$

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:  \( -\frac{3093415171201234}{21067389510348127} a^{19} + \frac{3995297262417847}{21067389510348127} a^{18} - \frac{44021027591093565}{21067389510348127} a^{17} + \frac{21630488549235081}{21067389510348127} a^{16} - \frac{810469326172870383}{42134779020696254} a^{15} + \frac{187571776414151801}{21067389510348127} a^{14} - \frac{1688190672287414635}{21067389510348127} a^{13} + \frac{42046313268310019}{21067389510348127} a^{12} - \frac{4398093258761834362}{21067389510348127} a^{11} + \frac{644388965637407971}{21067389510348127} a^{10} - \frac{11374290843856254467}{42134779020696254} a^{9} + \frac{1385732166659438583}{21067389510348127} a^{8} - \frac{5420221386005106934}{21067389510348127} a^{7} + \frac{1961491855001204440}{21067389510348127} a^{6} - \frac{2443665429439004539}{21067389510348127} a^{5} + \frac{1195357422029762367}{21067389510348127} a^{4} - \frac{920590226717647992}{21067389510348127} a^{3} + \frac{354314839878220090}{21067389510348127} a^{2} - \frac{66113244307783934}{21067389510348127} a + \frac{31015556579292239}{42134779020696254} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 140644.599182 \)
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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.0.162778775259375.1, 10.0.52089208083.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
5Data not computed
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$