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Properties

Label 20.0.20038303066...000.22
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{34}\cdot 11^{18}$
Root discriminant $462.48$
Ramified primes $2, 3, 5, 11$
Class number $1082688640$ (GRH)
Class group $[2, 2, 2, 22, 22, 279620]$ (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![16134802744225, 0, 27032738744000, 0, 10579393605000, 0, 1804275074250, 0, 164473920600, 0, 8841854010, 0, 292248825, 0, 5940000, 0, 70950, 0, 440, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 440*x^18 + 70950*x^16 + 5940000*x^14 + 292248825*x^12 + 8841854010*x^10 + 164473920600*x^8 + 1804275074250*x^6 + 10579393605000*x^4 + 27032738744000*x^2 + 16134802744225)
 
gp: K = bnfinit(x^20 + 440*x^18 + 70950*x^16 + 5940000*x^14 + 292248825*x^12 + 8841854010*x^10 + 164473920600*x^8 + 1804275074250*x^6 + 10579393605000*x^4 + 27032738744000*x^2 + 16134802744225, 1)
 

Normalized defining polynomial

\( x^{20} + 440 x^{18} + 70950 x^{16} + 5940000 x^{14} + 292248825 x^{12} + 8841854010 x^{10} + 164473920600 x^{8} + 1804275074250 x^{6} + 10579393605000 x^{4} + 27032738744000 x^{2} + 16134802744225 \)

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:  \(200383030666749741651348266601562500000000000000000000=2^{20}\cdot 3^{10}\cdot 5^{34}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $462.48$
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:  \(3300=2^{2}\cdot 3\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3300}(1,·)$, $\chi_{3300}(1159,·)$, $\chi_{3300}(3019,·)$, $\chi_{3300}(3101,·)$, $\chi_{3300}(1421,·)$, $\chi_{3300}(1361,·)$, $\chi_{3300}(1939,·)$, $\chi_{3300}(1879,·)$, $\chi_{3300}(281,·)$, $\chi_{3300}(3119,·)$, $\chi_{3300}(2141,·)$, $\chi_{3300}(2279,·)$, $\chi_{3300}(3299,·)$, $\chi_{3300}(359,·)$, $\chi_{3300}(199,·)$, $\chi_{3300}(239,·)$, $\chi_{3300}(3061,·)$, $\chi_{3300}(1021,·)$, $\chi_{3300}(2941,·)$, $\chi_{3300}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{6} - \frac{1}{9}$, $\frac{1}{9} a^{7} - \frac{1}{9} a$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{3}$, $\frac{1}{5940} a^{10} - \frac{1}{54} a^{8} - \frac{5}{108} a^{6} + \frac{2}{27} a^{4} - \frac{4}{27} a^{2} - \frac{13}{108}$, $\frac{1}{5940} a^{11} - \frac{1}{54} a^{9} - \frac{5}{108} a^{7} + \frac{2}{27} a^{5} - \frac{4}{27} a^{3} - \frac{13}{108} a$, $\frac{1}{17820} a^{12} - \frac{1}{36} a^{8} - \frac{1}{162} a^{6} - \frac{17}{36} a^{2} - \frac{13}{162}$, $\frac{1}{17820} a^{13} - \frac{1}{36} a^{9} - \frac{1}{162} a^{7} - \frac{17}{36} a^{3} - \frac{13}{162} a$, $\frac{1}{53460} a^{14} - \frac{1}{53460} a^{12} + \frac{1}{17820} a^{10} + \frac{19}{972} a^{8} - \frac{10}{243} a^{6} + \frac{17}{324} a^{4} - \frac{317}{972} a^{2} - \frac{91}{243}$, $\frac{1}{53460} a^{15} - \frac{1}{53460} a^{13} + \frac{1}{17820} a^{11} + \frac{19}{972} a^{9} - \frac{10}{243} a^{7} + \frac{17}{324} a^{5} - \frac{317}{972} a^{3} - \frac{91}{243} a$, $\frac{1}{160380} a^{16} + \frac{1}{160380} a^{14} + \frac{1}{160380} a^{12} - \frac{1}{80190} a^{10} - \frac{14}{729} a^{8} - \frac{14}{729} a^{6} + \frac{109}{2916} a^{4} - \frac{229}{1458} a^{2} - \frac{1187}{2916}$, $\frac{1}{160380} a^{17} + \frac{1}{160380} a^{15} + \frac{1}{160380} a^{13} - \frac{1}{80190} a^{11} - \frac{14}{729} a^{9} - \frac{14}{729} a^{7} + \frac{109}{2916} a^{5} - \frac{229}{1458} a^{3} - \frac{1187}{2916} a$, $\frac{1}{8788363487959629652802683654860} a^{18} - \frac{952998680523484152044059}{439418174397981482640134182743} a^{16} - \frac{11828867211907786657889783}{8788363487959629652802683654860} a^{14} - \frac{234322749917926602009439}{72035766294751062727890849630} a^{12} + \frac{33161594873098947808866955}{878836348795962965280268365486} a^{10} - \frac{1383916518653030527148182613}{159788427053811448232776066452} a^{8} + \frac{2012439752500329883427574526}{39947106763452862058194016613} a^{6} + \frac{2428084206175338341053299916}{39947106763452862058194016613} a^{4} - \frac{10758275580749583090537314552}{39947106763452862058194016613} a^{2} - \frac{118524485000507931398938621}{8877134836322858235154225914}$, $\frac{1}{7060246056777711956764522349019294180} a^{19} - \frac{1612367389889918884672455365689}{641840550616155632433138395365390380} a^{17} - \frac{2665300493595954060656064994181}{320920275308077816216569197682695190} a^{15} + \frac{3572649934329812348619230755}{2104395247921821745682420968411116} a^{13} - \frac{10347436382373467755053790877251}{160460137654038908108284598841347595} a^{11} + \frac{3961702461193460587563490517053225}{128368110123231126486627679073078076} a^{9} - \frac{18847125048146023727290337080693}{2917457048255252874696083615297229} a^{7} - \frac{524689748509572101335665195508513}{11669828193021011498784334461188916} a^{5} - \frac{977210701308843026601279740679637}{2917457048255252874696083615297229} a^{3} + \frac{408810680383807703966523676702811}{3889942731007003832928111487062972} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{22}\times C_{22}\times C_{279620}$, which has order $1082688640$ (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:  \( 65307151733.14945 \) (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

\(\Q(\sqrt{-165}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-5}, \sqrt{33})\), 5.5.5719140625.2, 10.0.447641631963281250000000000.1, 10.10.87430006242828369140625.1, 10.0.167467875781250000000000.4

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
5Data not computed
11Data not computed