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Properties

Label 20.4.82802905234...3125.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{10}\cdot 5^{15}\cdot 11^{16}$
Root discriminant $39.44$
Ramified primes $3, 5, 11$
Class number $5$ (GRH)
Class group $[5]$ (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![991, 22468, 46712, 9920, -101396, -350538, -574374, -467528, -210554, -35474, 20819, 12630, 2428, -294, 358, 148, 42, -40, -6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 6*x^18 - 40*x^17 + 42*x^16 + 148*x^15 + 358*x^14 - 294*x^13 + 2428*x^12 + 12630*x^11 + 20819*x^10 - 35474*x^9 - 210554*x^8 - 467528*x^7 - 574374*x^6 - 350538*x^5 - 101396*x^4 + 9920*x^3 + 46712*x^2 + 22468*x + 991)
 
gp: K = bnfinit(x^20 - 2*x^19 - 6*x^18 - 40*x^17 + 42*x^16 + 148*x^15 + 358*x^14 - 294*x^13 + 2428*x^12 + 12630*x^11 + 20819*x^10 - 35474*x^9 - 210554*x^8 - 467528*x^7 - 574374*x^6 - 350538*x^5 - 101396*x^4 + 9920*x^3 + 46712*x^2 + 22468*x + 991, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 6 x^{18} - 40 x^{17} + 42 x^{16} + 148 x^{15} + 358 x^{14} - 294 x^{13} + 2428 x^{12} + 12630 x^{11} + 20819 x^{10} - 35474 x^{9} - 210554 x^{8} - 467528 x^{7} - 574374 x^{6} - 350538 x^{5} - 101396 x^{4} + 9920 x^{3} + 46712 x^{2} + 22468 x + 991 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(82802905234194108120391845703125=3^{10}\cdot 5^{15}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.44$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{10} a^{12} + \frac{1}{5} a^{11} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{3}{10} a^{7} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{10} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{10} a^{8} - \frac{2}{5} a^{7} - \frac{1}{2} a^{6} + \frac{1}{5} a^{5} + \frac{1}{10} a^{3} - \frac{2}{5} a^{2} + \frac{3}{10} a - \frac{1}{5}$, $\frac{1}{10} a^{14} + \frac{1}{10} a^{11} + \frac{1}{5} a^{10} + \frac{3}{10} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{3}{10} a^{6} - \frac{1}{10} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{10} a + \frac{2}{5}$, $\frac{1}{10} a^{15} - \frac{1}{5} a^{10} + \frac{2}{5} a^{5} - \frac{1}{10}$, $\frac{1}{50} a^{16} - \frac{1}{25} a^{13} + \frac{1}{25} a^{12} - \frac{1}{5} a^{11} - \frac{1}{50} a^{10} - \frac{4}{25} a^{9} + \frac{2}{25} a^{8} + \frac{6}{25} a^{7} + \frac{12}{25} a^{6} + \frac{11}{50} a^{5} + \frac{12}{25} a^{4} + \frac{8}{25} a^{3} + \frac{3}{25} a^{2} + \frac{11}{50} a - \frac{19}{50}$, $\frac{1}{50} a^{17} - \frac{1}{25} a^{14} + \frac{1}{25} a^{13} - \frac{3}{25} a^{11} - \frac{4}{25} a^{10} - \frac{8}{25} a^{9} - \frac{4}{25} a^{8} - \frac{3}{25} a^{7} - \frac{7}{25} a^{6} + \frac{12}{25} a^{5} - \frac{7}{25} a^{4} - \frac{2}{25} a^{3} + \frac{1}{50} a^{2} - \frac{2}{25} a + \frac{1}{5}$, $\frac{1}{264250} a^{18} - \frac{1311}{264250} a^{17} - \frac{871}{264250} a^{16} + \frac{4639}{132125} a^{15} - \frac{5083}{132125} a^{14} - \frac{172}{5285} a^{13} - \frac{3534}{132125} a^{12} - \frac{257}{264250} a^{11} + \frac{593}{264250} a^{10} - \frac{17517}{132125} a^{9} - \frac{12151}{132125} a^{8} + \frac{5667}{26425} a^{7} - \frac{55281}{264250} a^{6} + \frac{7521}{264250} a^{5} + \frac{38198}{132125} a^{4} - \frac{92971}{264250} a^{3} + \frac{9807}{37750} a^{2} - \frac{50801}{132125} a - \frac{881}{264250}$, $\frac{1}{1810184281196593180197275153043372422380750} a^{19} + \frac{168922438175117889834757939861802548}{129298877228328084299805368074526601598625} a^{18} - \frac{726201277529356489704448626893198494361}{129298877228328084299805368074526601598625} a^{17} - \frac{78357068770495097721646079635070339258}{36203685623931863603945503060867448447615} a^{16} - \frac{29292277307854105404921042994032392797521}{905092140598296590098637576521686211190375} a^{15} + \frac{30202820038238113684194572525290478447981}{905092140598296590098637576521686211190375} a^{14} - \frac{37408616866905852938474094306136171017813}{1810184281196593180197275153043372422380750} a^{13} - \frac{11831224651797955148223974726642172981871}{1810184281196593180197275153043372422380750} a^{12} - \frac{436766984709534581620800164045961619928643}{1810184281196593180197275153043372422380750} a^{11} - \frac{6119794199247582220521347128858546105602}{181018428119659318019727515304337242238075} a^{10} + \frac{286883339288148285977096403506265013331988}{905092140598296590098637576521686211190375} a^{9} + \frac{564029035144830568516444591031638349926899}{1810184281196593180197275153043372422380750} a^{8} + \frac{827279249570464260611412237857163575830779}{1810184281196593180197275153043372422380750} a^{7} - \frac{408229011504823735419498141419716215093537}{1810184281196593180197275153043372422380750} a^{6} - \frac{209495504783088027306201099871479796472438}{905092140598296590098637576521686211190375} a^{5} - \frac{47696780038982958453115247249029993590913}{1810184281196593180197275153043372422380750} a^{4} - \frac{348836750996080526330312495055223576398549}{1810184281196593180197275153043372422380750} a^{3} - \frac{142644593321417535232983820490443706564923}{362036856239318636039455030608674484476150} a^{2} - \frac{62467992918099883699832900825839161209111}{258597754456656168599610736149053203197250} a + \frac{75069948668814475529029995070697099002621}{905092140598296590098637576521686211190375}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  $11$
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:  \( 32045543.871448386 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 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 ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$