↑

Properties

Label 20.0.17910060025...4976.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 7^{14}\cdot 199^{8}$
Root discriminant $91.76$
Ramified primes $2, 7, 199$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_5\times A_4$ (as 20T37)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![58471, -54390, 126562, -60510, 44861, -27568, 23172, -7864, 9099, 820, 5254, -8874, -57, 2726, 466, -876, 34, 104, -10, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 10*x^18 + 104*x^17 + 34*x^16 - 876*x^15 + 466*x^14 + 2726*x^13 - 57*x^12 - 8874*x^11 + 5254*x^10 + 820*x^9 + 9099*x^8 - 7864*x^7 + 23172*x^6 - 27568*x^5 + 44861*x^4 - 60510*x^3 + 126562*x^2 - 54390*x + 58471)
 
gp: K = bnfinit(x^20 - 6*x^19 - 10*x^18 + 104*x^17 + 34*x^16 - 876*x^15 + 466*x^14 + 2726*x^13 - 57*x^12 - 8874*x^11 + 5254*x^10 + 820*x^9 + 9099*x^8 - 7864*x^7 + 23172*x^6 - 27568*x^5 + 44861*x^4 - 60510*x^3 + 126562*x^2 - 54390*x + 58471, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 10 x^{18} + 104 x^{17} + 34 x^{16} - 876 x^{15} + 466 x^{14} + 2726 x^{13} - 57 x^{12} - 8874 x^{11} + 5254 x^{10} + 820 x^{9} + 9099 x^{8} - 7864 x^{7} + 23172 x^{6} - 27568 x^{5} + 44861 x^{4} - 60510 x^{3} + 126562 x^{2} - 54390 x + 58471 \)

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:  \(1791006002586548906938780335646468734976=2^{30}\cdot 7^{14}\cdot 199^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.76$
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:  $2, 7, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} - \frac{1}{7} a^{12} + \frac{2}{7} a^{11} + \frac{3}{7} a^{10} - \frac{3}{7} a^{9} - \frac{1}{7} a^{8} + \frac{3}{7} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2}$, $\frac{1}{7} a^{14} + \frac{1}{7} a^{12} - \frac{2}{7} a^{11} + \frac{3}{7} a^{9} + \frac{2}{7} a^{8} + \frac{3}{7} a^{6} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2}$, $\frac{1}{7} a^{15} - \frac{1}{7} a^{12} - \frac{2}{7} a^{11} - \frac{2}{7} a^{9} + \frac{1}{7} a^{8} + \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{3}{7} a^{3} - \frac{2}{7} a^{2}$, $\frac{1}{7} a^{16} - \frac{3}{7} a^{12} + \frac{2}{7} a^{11} + \frac{1}{7} a^{10} - \frac{2}{7} a^{9} - \frac{1}{7} a^{8} - \frac{1}{7} a^{7} - \frac{2}{7} a^{6} - \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{3}{7} a^{3} + \frac{2}{7} a^{2}$, $\frac{1}{7} a^{17} - \frac{1}{7} a^{12} - \frac{3}{7} a^{9} + \frac{3}{7} a^{8} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} - \frac{1}{7} a^{3} - \frac{1}{7} a^{2}$, $\frac{1}{91} a^{18} + \frac{6}{91} a^{17} + \frac{3}{91} a^{16} - \frac{6}{91} a^{15} + \frac{6}{91} a^{14} + \frac{5}{91} a^{13} - \frac{23}{91} a^{12} - \frac{3}{91} a^{11} - \frac{10}{91} a^{10} - \frac{16}{91} a^{9} + \frac{36}{91} a^{8} + \frac{12}{91} a^{7} + \frac{8}{91} a^{6} + \frac{2}{7} a^{5} - \frac{11}{91} a^{4} + \frac{29}{91} a^{3} - \frac{6}{91} a^{2} + \frac{6}{13} a + \frac{1}{13}$, $\frac{1}{2465283208072135442757408382514138449897604429} a^{19} - \frac{1178738245931595728806306265841708702722488}{352183315438876491822486911787734064271086347} a^{18} - \frac{115184614250684064227820969640976482569322326}{2465283208072135442757408382514138449897604429} a^{17} - \frac{139102257045184267836353731204587763704445888}{2465283208072135442757408382514138449897604429} a^{16} + \frac{62590280276573573276442841901160458081157953}{2465283208072135442757408382514138449897604429} a^{15} - \frac{6163796472726358212854326140238130318696438}{189637169851702726365954490962626034607508033} a^{14} - \frac{168824148763886203273711672333557540200568868}{2465283208072135442757408382514138449897604429} a^{13} + \frac{714600171709458919226712311570577811867909997}{2465283208072135442757408382514138449897604429} a^{12} - \frac{322488139508328645059366225900110541293682185}{2465283208072135442757408382514138449897604429} a^{11} + \frac{43773965110965698292486169767020554130055548}{2465283208072135442757408382514138449897604429} a^{10} - \frac{110529389189590201802343560365151795884688327}{352183315438876491822486911787734064271086347} a^{9} + \frac{223041134482251375925967617535554549976128861}{2465283208072135442757408382514138449897604429} a^{8} + \frac{881535117955513305641427787255661128765476079}{2465283208072135442757408382514138449897604429} a^{7} - \frac{1017765543742427715567015050774032628785260528}{2465283208072135442757408382514138449897604429} a^{6} - \frac{991743128754278784041926643600204309341402447}{2465283208072135442757408382514138449897604429} a^{5} - \frac{714014625657237243880698115597029660060072804}{2465283208072135442757408382514138449897604429} a^{4} - \frac{144637026163056622865745661906759877394699719}{2465283208072135442757408382514138449897604429} a^{3} + \frac{807092338194157265443234961612193573862124876}{2465283208072135442757408382514138449897604429} a^{2} - \frac{86629055154933294283348570745234481628571736}{352183315438876491822486911787734064271086347} a + \frac{156229639770909003919745169646692225955987190}{352183315438876491822486911787734064271086347}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 483814559041 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\times A_4$ (as 20T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 16 conjugacy class representatives for $D_5\times A_4$
Character table for $D_5\times A_4$

Intermediate fields

4.0.3136.1, 5.5.1940449.1

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

Sibling fields

Degree 30 siblings: data not computed
Degree 40 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 R $15{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ $15{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{5}$ $15{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
2Data not computed
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
199Data not computed