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Properties

Label 20.0.37603126978...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{16}\cdot 5^{15}\cdot 17^{15}$
Root discriminant $67.42$
Ramified primes $3, 5, 17$
Class number $20$ (GRH)
Class group $[20]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![132281, 109286, -105289, -127378, -86572, 41942, 181783, -2255, 55855, -85065, 18813, -9645, 16595, -127, 1703, -1916, 427, -80, 31, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 31*x^18 - 80*x^17 + 427*x^16 - 1916*x^15 + 1703*x^14 - 127*x^13 + 16595*x^12 - 9645*x^11 + 18813*x^10 - 85065*x^9 + 55855*x^8 - 2255*x^7 + 181783*x^6 + 41942*x^5 - 86572*x^4 - 127378*x^3 - 105289*x^2 + 109286*x + 132281)
 
gp: K = bnfinit(x^20 - 5*x^19 + 31*x^18 - 80*x^17 + 427*x^16 - 1916*x^15 + 1703*x^14 - 127*x^13 + 16595*x^12 - 9645*x^11 + 18813*x^10 - 85065*x^9 + 55855*x^8 - 2255*x^7 + 181783*x^6 + 41942*x^5 - 86572*x^4 - 127378*x^3 - 105289*x^2 + 109286*x + 132281, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 31 x^{18} - 80 x^{17} + 427 x^{16} - 1916 x^{15} + 1703 x^{14} - 127 x^{13} + 16595 x^{12} - 9645 x^{11} + 18813 x^{10} - 85065 x^{9} + 55855 x^{8} - 2255 x^{7} + 181783 x^{6} + 41942 x^{5} - 86572 x^{4} - 127378 x^{3} - 105289 x^{2} + 109286 x + 132281 \)

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:  \(3760312697824452795491478057861328125=3^{16}\cdot 5^{15}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.42$
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:  $3, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{2}{9} a^{9} - \frac{2}{9} a^{8} - \frac{1}{9} a^{7} + \frac{4}{9} a^{6} - \frac{4}{9} a^{5} - \frac{1}{9} a^{4} - \frac{1}{9} a^{3} - \frac{1}{9} a^{2} + \frac{2}{9}$, $\frac{1}{18} a^{14} - \frac{1}{18} a^{13} + \frac{1}{18} a^{12} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{18} a^{8} - \frac{7}{18} a^{7} + \frac{1}{18} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} + \frac{1}{18} a^{2} + \frac{5}{18} a + \frac{7}{18}$, $\frac{1}{54} a^{15} - \frac{1}{54} a^{14} - \frac{1}{54} a^{13} - \frac{1}{18} a^{12} + \frac{7}{54} a^{11} + \frac{1}{54} a^{10} + \frac{17}{54} a^{9} - \frac{5}{18} a^{8} + \frac{5}{18} a^{7} - \frac{23}{54} a^{6} + \frac{23}{54} a^{5} + \frac{17}{54} a^{4} - \frac{1}{2} a^{3} + \frac{19}{54} a^{2} - \frac{23}{54} a - \frac{5}{27}$, $\frac{1}{54} a^{16} + \frac{1}{54} a^{14} - \frac{1}{54} a^{13} + \frac{7}{54} a^{12} + \frac{5}{54} a^{11} - \frac{1}{18} a^{10} - \frac{19}{54} a^{9} - \frac{1}{2} a^{8} + \frac{19}{54} a^{7} + \frac{1}{6} a^{6} + \frac{7}{54} a^{5} - \frac{25}{54} a^{4} - \frac{5}{54} a^{3} + \frac{11}{54} a^{2} - \frac{1}{3} a - \frac{13}{54}$, $\frac{1}{162} a^{17} + \frac{1}{162} a^{16} + \frac{1}{162} a^{15} + \frac{1}{27} a^{13} + \frac{2}{27} a^{12} - \frac{8}{81} a^{11} + \frac{7}{81} a^{10} - \frac{32}{81} a^{9} + \frac{14}{81} a^{8} + \frac{32}{81} a^{7} - \frac{1}{81} a^{6} - \frac{1}{3} a^{5} + \frac{7}{27} a^{4} + \frac{13}{27} a^{3} + \frac{11}{162} a^{2} + \frac{41}{162} a - \frac{49}{162}$, $\frac{1}{1763694} a^{18} + \frac{559}{881847} a^{17} + \frac{1174}{881847} a^{16} + \frac{8515}{1763694} a^{15} + \frac{2336}{293949} a^{14} + \frac{9596}{293949} a^{13} + \frac{52057}{881847} a^{12} + \frac{24503}{881847} a^{11} + \frac{34004}{881847} a^{10} - \frac{113245}{293949} a^{9} + \frac{190198}{881847} a^{8} - \frac{428339}{881847} a^{7} - \frac{139438}{881847} a^{6} + \frac{48752}{293949} a^{5} - \frac{17870}{293949} a^{4} - \frac{16969}{92826} a^{3} + \frac{226022}{881847} a^{2} + \frac{147758}{881847} a + \frac{537821}{1763694}$, $\frac{1}{682248002511778697161090517490579580356509918658} a^{19} + \frac{55789708699598081659475266880611961697}{935868316202714262223718130988449355770246802} a^{18} - \frac{431866841677584885716771756915080457215917389}{682248002511778697161090517490579580356509918658} a^{17} + \frac{992659807794834880221891975829307634719115781}{227416000837259565720363505830193193452169972886} a^{16} + \frac{168375663434954257129412040805331606335922720}{341124001255889348580545258745289790178254959329} a^{15} - \frac{2840094798902705560723913284806384885532128932}{113708000418629782860181752915096596726084986443} a^{14} + \frac{2448595866854993201316401119760182290033698524}{48732000179412764082935036963612827168322137047} a^{13} + \frac{10745038822538925310733938159839903287026018717}{113708000418629782860181752915096596726084986443} a^{12} + \frac{47720274046343420904184624775754944631961195081}{341124001255889348580545258745289790178254959329} a^{11} + \frac{21035003209186583462831303783855501145951428168}{341124001255889348580545258745289790178254959329} a^{10} - \frac{167690065346977910302832242755622237846175185832}{341124001255889348580545258745289790178254959329} a^{9} - \frac{166592992446788769385527494729687118572008156837}{341124001255889348580545258745289790178254959329} a^{8} - \frac{40350372623117546613168134495644734930334076771}{113708000418629782860181752915096596726084986443} a^{7} - \frac{57261623267343441955463177768007654641805569251}{341124001255889348580545258745289790178254959329} a^{6} + \frac{3631889502526083077199901265303070553012770178}{37902666806209927620060584305032198908694995481} a^{5} + \frac{1514324219940149197229540713421111489474157101}{5129684229411869903466845996169771280876014426} a^{4} + \frac{10817233615633335478554647768903843214959238091}{75805333612419855240121168610064397817389990962} a^{3} + \frac{339567091889821551109592072349361467688082160017}{682248002511778697161090517490579580356509918658} a^{2} + \frac{23749657515621163434431056389851647669541209}{1154395943336342973199814750407072047980558238} a + \frac{4427634042611373057086852885002053435541319287}{17953894802941544662133960986594199483066050491}$

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

Class group and class number

$C_{20}$, which has order $20$ (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:  \( 5918089885.052663 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{85}) \), 4.0.614125.1, 5.1.2926125.1, 10.2.727787638828125.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
5Data not computed
17Data not computed