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Properties

Label 20.20.3409580537...3125.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{18}\cdot 5^{15}\cdot 7^{8}\cdot 29^{8}$
Root discriminant $75.27$
Ramified primes $3, 5, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_5:F_5$ (as 20T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-707225, 3018000, 1728925, -10277550, -747660, 13024230, -526425, -8513820, 616446, 3212694, -273903, -730512, 71988, 100758, -11667, -8142, 1098, 348, -53, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 53*x^18 + 348*x^17 + 1098*x^16 - 8142*x^15 - 11667*x^14 + 100758*x^13 + 71988*x^12 - 730512*x^11 - 273903*x^10 + 3212694*x^9 + 616446*x^8 - 8513820*x^7 - 526425*x^6 + 13024230*x^5 - 747660*x^4 - 10277550*x^3 + 1728925*x^2 + 3018000*x - 707225)
 
gp: K = bnfinit(x^20 - 6*x^19 - 53*x^18 + 348*x^17 + 1098*x^16 - 8142*x^15 - 11667*x^14 + 100758*x^13 + 71988*x^12 - 730512*x^11 - 273903*x^10 + 3212694*x^9 + 616446*x^8 - 8513820*x^7 - 526425*x^6 + 13024230*x^5 - 747660*x^4 - 10277550*x^3 + 1728925*x^2 + 3018000*x - 707225, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 53 x^{18} + 348 x^{17} + 1098 x^{16} - 8142 x^{15} - 11667 x^{14} + 100758 x^{13} + 71988 x^{12} - 730512 x^{11} - 273903 x^{10} + 3212694 x^{9} + 616446 x^{8} - 8513820 x^{7} - 526425 x^{6} + 13024230 x^{5} - 747660 x^{4} - 10277550 x^{3} + 1728925 x^{2} + 3018000 x - 707225 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(34095805371377238983021762969970703125=3^{18}\cdot 5^{15}\cdot 7^{8}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.27$
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:  $3, 5, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{20} a^{10} - \frac{3}{20} a^{9} + \frac{1}{5} a^{8} + \frac{1}{20} a^{7} + \frac{1}{5} a^{6} - \frac{3}{20} a^{5} + \frac{3}{10} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{20} a^{11} - \frac{1}{4} a^{9} + \frac{3}{20} a^{8} - \frac{3}{20} a^{7} - \frac{1}{20} a^{6} - \frac{3}{20} a^{5} - \frac{7}{20} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{20} a^{12} - \frac{1}{10} a^{9} - \frac{3}{20} a^{8} + \frac{1}{5} a^{7} - \frac{3}{20} a^{6} - \frac{1}{10} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{20} a^{13} + \frac{1}{20} a^{9} + \frac{1}{10} a^{8} - \frac{1}{20} a^{7} - \frac{1}{5} a^{6} - \frac{1}{20} a^{5} + \frac{1}{10} a^{4} - \frac{1}{4} a$, $\frac{1}{40} a^{14} - \frac{1}{40} a^{11} - \frac{1}{40} a^{10} - \frac{7}{40} a^{9} - \frac{1}{20} a^{8} - \frac{3}{40} a^{7} + \frac{1}{20} a^{6} + \frac{1}{40} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{200} a^{15} - \frac{1}{100} a^{14} - \frac{3}{200} a^{12} - \frac{1}{200} a^{11} + \frac{1}{200} a^{10} + \frac{6}{25} a^{9} - \frac{3}{40} a^{8} - \frac{6}{25} a^{7} + \frac{49}{200} a^{6} + \frac{19}{40} a^{5} - \frac{17}{40} a^{4} + \frac{2}{5} a^{3} - \frac{1}{10} a^{2} + \frac{1}{8} a$, $\frac{1}{200} a^{16} + \frac{1}{200} a^{14} - \frac{3}{200} a^{13} + \frac{3}{200} a^{12} + \frac{1}{50} a^{11} - \frac{1}{40} a^{10} + \frac{13}{100} a^{9} + \frac{3}{50} a^{8} - \frac{1}{100} a^{7} - \frac{37}{200} a^{6} + \frac{1}{20} a^{5} + \frac{3}{40} a^{4} + \frac{13}{40} a^{3} - \frac{3}{40} a^{2} + \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{200} a^{17} - \frac{1}{200} a^{14} + \frac{3}{200} a^{13} - \frac{3}{200} a^{12} - \frac{1}{50} a^{11} - \frac{1}{40} a^{10} - \frac{13}{100} a^{9} + \frac{23}{200} a^{8} + \frac{41}{200} a^{7} - \frac{29}{200} a^{6} + \frac{3}{20} a^{5} + \frac{1}{10} a^{4} + \frac{11}{40} a^{3} - \frac{3}{20} a^{2} + \frac{1}{4} a$, $\frac{1}{400} a^{18} - \frac{1}{400} a^{17} - \frac{1}{400} a^{16} + \frac{1}{400} a^{14} + \frac{7}{400} a^{13} - \frac{7}{400} a^{12} + \frac{1}{100} a^{11} - \frac{1}{80} a^{10} + \frac{1}{400} a^{9} + \frac{81}{400} a^{8} - \frac{23}{200} a^{7} + \frac{7}{80} a^{6} + \frac{21}{80} a^{5} - \frac{11}{80} a^{4} + \frac{13}{40} a^{3} - \frac{7}{16} a^{2} - \frac{3}{16} a - \frac{1}{16}$, $\frac{1}{2030509518045434835887600} a^{19} - \frac{142193473337047681091}{406101903609086967177520} a^{18} + \frac{1715122890324482746027}{2030509518045434835887600} a^{17} + \frac{365489507756609850841}{253813689755679354485950} a^{16} - \frac{2781203326925492300027}{2030509518045434835887600} a^{15} - \frac{15433017709068968764101}{2030509518045434835887600} a^{14} - \frac{3053175745511944265081}{406101903609086967177520} a^{13} + \frac{4310131453307373424043}{1015254759022717417943800} a^{12} - \frac{22738658122768994404003}{2030509518045434835887600} a^{11} + \frac{27435514517837981613253}{2030509518045434835887600} a^{10} - \frac{29266402890465287238827}{2030509518045434835887600} a^{9} - \frac{19447378623747171185179}{203050951804543483588760} a^{8} - \frac{70145077750547362726821}{2030509518045434835887600} a^{7} + \frac{325313089188589545172339}{2030509518045434835887600} a^{6} + \frac{153460049027724495715517}{406101903609086967177520} a^{5} - \frac{1985292410479475646703}{25381368975567935448595} a^{4} + \frac{80547467164778177727439}{406101903609086967177520} a^{3} + \frac{91681717493128114562641}{406101903609086967177520} a^{2} + \frac{19401861049389830090513}{81220380721817393435504} a - \frac{4433828340489022246813}{40610190360908696717752}$

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:  $19$
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:  \( 8111181538130 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_5:F_5$ (as 20T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 20 conjugacy class representatives for $C_2\times C_5:F_5$
Character table for $C_2\times C_5:F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 10.10.870450781956328125.1

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 R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
3Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.5.4.1$x^{5} - 29$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
29.5.4.1$x^{5} - 29$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$