↑

Properties

Label 20.20.1856327181...8125.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{16}\cdot 5^{15}\cdot 7^{10}\cdot 29^{8}$
Root discriminant $81.93$
Ramified primes $3, 5, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5:F_5$ (as 20T27)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-727355, 6935985, -8641610, -30429375, 32146026, 47209428, -31272909, -31935498, 14146635, 11214798, -3537582, -2205189, 530238, 247119, -49362, -15396, 2769, 486, -83, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 83*x^18 + 486*x^17 + 2769*x^16 - 15396*x^15 - 49362*x^14 + 247119*x^13 + 530238*x^12 - 2205189*x^11 - 3537582*x^10 + 11214798*x^9 + 14146635*x^8 - 31935498*x^7 - 31272909*x^6 + 47209428*x^5 + 32146026*x^4 - 30429375*x^3 - 8641610*x^2 + 6935985*x - 727355)
 
gp: K = bnfinit(x^20 - 6*x^19 - 83*x^18 + 486*x^17 + 2769*x^16 - 15396*x^15 - 49362*x^14 + 247119*x^13 + 530238*x^12 - 2205189*x^11 - 3537582*x^10 + 11214798*x^9 + 14146635*x^8 - 31935498*x^7 - 31272909*x^6 + 47209428*x^5 + 32146026*x^4 - 30429375*x^3 - 8641610*x^2 + 6935985*x - 727355, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 83 x^{18} + 486 x^{17} + 2769 x^{16} - 15396 x^{15} - 49362 x^{14} + 247119 x^{13} + 530238 x^{12} - 2205189 x^{11} - 3537582 x^{10} + 11214798 x^{9} + 14146635 x^{8} - 31935498 x^{7} - 31272909 x^{6} + 47209428 x^{5} + 32146026 x^{4} - 30429375 x^{3} - 8641610 x^{2} + 6935985 x - 727355 \)

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:  \(185632718133053856685340709503173828125=3^{16}\cdot 5^{15}\cdot 7^{10}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.93$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{8} + \frac{1}{6} a^{6} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} + \frac{1}{6} a^{7} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{13} - \frac{1}{12} a^{10} - \frac{1}{2} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{12} a^{4} + \frac{1}{4} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{12}$, $\frac{1}{348} a^{15} - \frac{5}{174} a^{14} - \frac{1}{116} a^{13} + \frac{7}{174} a^{12} - \frac{11}{348} a^{11} - \frac{1}{348} a^{10} + \frac{7}{87} a^{9} + \frac{1}{174} a^{8} + \frac{31}{87} a^{7} - \frac{5}{58} a^{6} + \frac{139}{348} a^{5} - \frac{41}{174} a^{4} + \frac{23}{116} a^{3} + \frac{17}{58} a^{2} - \frac{167}{348} a + \frac{169}{348}$, $\frac{1}{1740} a^{16} + \frac{1}{1740} a^{15} - \frac{11}{348} a^{14} - \frac{9}{116} a^{13} - \frac{89}{1740} a^{12} + \frac{11}{174} a^{11} - \frac{41}{1740} a^{10} - \frac{9}{58} a^{9} + \frac{73}{870} a^{8} - \frac{1}{3} a^{7} - \frac{73}{348} a^{6} - \frac{409}{1740} a^{5} + \frac{167}{580} a^{4} - \frac{83}{348} a^{3} - \frac{33}{116} a^{2} + \frac{71}{174} a + \frac{35}{116}$, $\frac{1}{1740} a^{17} - \frac{1}{1740} a^{15} - \frac{5}{174} a^{14} - \frac{119}{1740} a^{13} + \frac{33}{580} a^{12} + \frac{19}{290} a^{11} + \frac{1}{290} a^{10} + \frac{18}{145} a^{9} - \frac{154}{435} a^{8} - \frac{53}{116} a^{7} - \frac{89}{290} a^{6} - \frac{1}{12} a^{5} - \frac{131}{290} a^{4} + \frac{47}{348} a^{3} - \frac{5}{12} a^{2} + \frac{10}{87} a + \frac{6}{29}$, $\frac{1}{1740} a^{18} + \frac{1}{1740} a^{15} + \frac{17}{580} a^{14} - \frac{41}{1740} a^{13} - \frac{1}{12} a^{12} - \frac{12}{145} a^{11} - \frac{1}{87} a^{10} + \frac{56}{435} a^{9} - \frac{259}{1740} a^{8} - \frac{106}{435} a^{7} - \frac{85}{174} a^{6} - \frac{3}{116} a^{5} + \frac{29}{60} a^{4} + \frac{143}{348} a^{3} - \frac{83}{348} a^{2} + \frac{55}{174} a - \frac{37}{87}$, $\frac{1}{312492896668107630447358294864280133928044600} a^{19} - \frac{55348100024689418563485940189812855443383}{312492896668107630447358294864280133928044600} a^{18} + \frac{2787528303253613863415285688175359844817}{78123224167026907611839573716070033482011150} a^{17} + \frac{107224217970315305630232490436945624083}{538780856324323500771307404938414024013870} a^{16} + \frac{160974971605340844072119818389788998240519}{312492896668107630447358294864280133928044600} a^{15} + \frac{5376799278414141677744535545851057452432041}{312492896668107630447358294864280133928044600} a^{14} - \frac{11267405832512476241328362357258490798363209}{312492896668107630447358294864280133928044600} a^{13} - \frac{76508056314128294439057532790933268235697}{1905444491878705063703404236977317889805150} a^{12} + \frac{12730747410136523804419253192006283191469547}{156246448334053815223679147432140066964022300} a^{11} - \frac{3249767479152950392580012374472053839459869}{104164298889369210149119431621426711309348200} a^{10} + \frac{40588165935594130176457116882888461762137267}{312492896668107630447358294864280133928044600} a^{9} - \frac{15225163935342143919481361490731193456138127}{104164298889369210149119431621426711309348200} a^{8} - \frac{2936574178943708885314015617755052532585297}{13020537361171151268639928952678338913668525} a^{7} + \frac{16180586466074830586210676520553334597932896}{39061612083513453805919786858035016741005575} a^{6} - \frac{8335435499478641474340686073757608875103637}{20832859777873842029823886324285342261869640} a^{5} + \frac{37189163908581156136814746877707797608588843}{312492896668107630447358294864280133928044600} a^{4} - \frac{5375775112400766087588757810641167240990669}{20832859777873842029823886324285342261869640} a^{3} + \frac{1696032679019418597164995268191630736882893}{7812322416702690761183957371607003348201115} a^{2} + \frac{164466587292878751170391339372377120609341}{2083285977787384202982388632428534226186964} a + \frac{19611509902765820695080347264439074581125847}{62498579333621526089471658972856026785608920}$

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:  \( 22123034286000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:F_5$ (as 20T27):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 10 conjugacy class representatives for $C_5:F_5$
Character table for $C_5:F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.6125.1, 10.10.870450781956328125.1 x5

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 sibling: data not computed
Degree 25 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 R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$$[\ ]_{4}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$29$29.2.0.1$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}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.10.8.1$x^{10} - 899 x^{5} + 204363$$5$$2$$8$$D_5$$[\ ]_{5}^{2}$