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Properties

Label 20.0.10019151533...8125.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{15}\cdot 11^{18}$
Root discriminant $50.12$
Ramified primes $3, 5, 11$
Class number $40$ (GRH)
Class group $[2, 2, 10]$ (GRH)
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![377245, -1608255, 3643915, -5497910, 6400086, -6354447, 5598538, -4255273, 2707188, -1456697, 694684, -299394, 115483, -40207, 12796, -3530, 928, -204, 42, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 42*x^18 - 204*x^17 + 928*x^16 - 3530*x^15 + 12796*x^14 - 40207*x^13 + 115483*x^12 - 299394*x^11 + 694684*x^10 - 1456697*x^9 + 2707188*x^8 - 4255273*x^7 + 5598538*x^6 - 6354447*x^5 + 6400086*x^4 - 5497910*x^3 + 3643915*x^2 - 1608255*x + 377245)
 
gp: K = bnfinit(x^20 - 8*x^19 + 42*x^18 - 204*x^17 + 928*x^16 - 3530*x^15 + 12796*x^14 - 40207*x^13 + 115483*x^12 - 299394*x^11 + 694684*x^10 - 1456697*x^9 + 2707188*x^8 - 4255273*x^7 + 5598538*x^6 - 6354447*x^5 + 6400086*x^4 - 5497910*x^3 + 3643915*x^2 - 1608255*x + 377245, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 42 x^{18} - 204 x^{17} + 928 x^{16} - 3530 x^{15} + 12796 x^{14} - 40207 x^{13} + 115483 x^{12} - 299394 x^{11} + 694684 x^{10} - 1456697 x^{9} + 2707188 x^{8} - 4255273 x^{7} + 5598538 x^{6} - 6354447 x^{5} + 6400086 x^{4} - 5497910 x^{3} + 3643915 x^{2} - 1608255 x + 377245 \)

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:  \(10019151533337487082567413330078125=3^{10}\cdot 5^{15}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.12$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{15} - \frac{1}{4} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{3}{8} a^{9} - \frac{1}{2} a^{8} - \frac{3}{8} a^{7} + \frac{3}{8} a^{6} + \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{3}{8} a^{9} - \frac{3}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{3}{8} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{3}{8} a + \frac{3}{8}$, $\frac{1}{45641496} a^{18} - \frac{320209}{11410374} a^{17} - \frac{1924259}{45641496} a^{16} - \frac{1186490}{5705187} a^{15} - \frac{53135}{22820748} a^{14} + \frac{1725451}{45641496} a^{13} - \frac{1263603}{7606916} a^{12} + \frac{1261015}{11410374} a^{11} - \frac{1896199}{45641496} a^{10} + \frac{12951415}{45641496} a^{9} - \frac{889095}{3803458} a^{8} + \frac{1467191}{15213832} a^{7} + \frac{680155}{3803458} a^{6} + \frac{8814257}{45641496} a^{5} - \frac{2473047}{15213832} a^{4} - \frac{1200571}{5705187} a^{3} - \frac{11025023}{45641496} a^{2} - \frac{1512641}{45641496} a + \frac{324119}{1201092}$, $\frac{1}{14840430911649108954252935761889122385872390244664} a^{19} - \frac{30285756206310406094055437483159891669401}{7420215455824554477126467880944561192936195122332} a^{18} - \frac{763042909918713010183895220891232814085861376663}{14840430911649108954252935761889122385872390244664} a^{17} + \frac{237692256939166588722267430415646571430914432211}{7420215455824554477126467880944561192936195122332} a^{16} + \frac{4727125752488673908459604459468465314768997365}{24489159920213051079625306537770829019591403044} a^{15} + \frac{733529157337544744062166224364096915429150154911}{14840430911649108954252935761889122385872390244664} a^{14} + \frac{176752349596944297581179017092762019223750698395}{3710107727912277238563233940472280596468097561166} a^{13} + \frac{895754740837863997009773941393493666414248352049}{3710107727912277238563233940472280596468097561166} a^{12} - \frac{242463668087747273063747481243785077316170973591}{14840430911649108954252935761889122385872390244664} a^{11} + \frac{3538286140085427528431727661591556452691465980145}{14840430911649108954252935761889122385872390244664} a^{10} - \frac{15186821000488435975282690157017326846291093349}{73467479760639153238875919613312487058774209132} a^{9} - \frac{414032142220838741902787143377911012431830453101}{4946810303883036318084311920629707461957463414888} a^{8} + \frac{648186709128534668594560001571520778625907861209}{2473405151941518159042155960314853730978731707444} a^{7} + \frac{2903014389020547024653999475517549005386456228117}{14840430911649108954252935761889122385872390244664} a^{6} - \frac{4317316897503880665906548243878490996313805846331}{14840430911649108954252935761889122385872390244664} a^{5} + \frac{2881864361165155278449632546846650465382379561285}{7420215455824554477126467880944561192936195122332} a^{4} - \frac{1203666744261374773362673795828214358141031358489}{4946810303883036318084311920629707461957463414888} a^{3} - \frac{1875072788386066514545330738911873132980938786517}{4946810303883036318084311920629707461957463414888} a^{2} - \frac{8258487960580877880261995166784830490037612055}{32544804630809449461080999477827022776035943519} a + \frac{909411406273649723755631216785335315104364934}{5138653362759386757012789391235845701479359503}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{10}$, which has order $40$ (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:  \( 26826674.447924238 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.136125.2, 5.1.1830125.1, 10.2.16746787578125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.10.9.5$x^{10} - 8019$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.5$x^{10} - 8019$$10$$1$$9$$C_{10}$$[\ ]_{10}$