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Properties

Label 20.8.14728777159...3125.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{13}\cdot 97^{2}\cdot 1039^{4}\cdot 1049^{2}$
Root discriminant $36.17$
Ramified primes $5, 97, 1039, 1049$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1039

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![141791, 366509, -259484, -349102, 311280, 172477, -81531, -70159, 13527, 30738, -4294, 1094, 8088, 241, -2449, -527, 316, 94, -24, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 24*x^18 + 94*x^17 + 316*x^16 - 527*x^15 - 2449*x^14 + 241*x^13 + 8088*x^12 + 1094*x^11 - 4294*x^10 + 30738*x^9 + 13527*x^8 - 70159*x^7 - 81531*x^6 + 172477*x^5 + 311280*x^4 - 349102*x^3 - 259484*x^2 + 366509*x + 141791)
 
gp: K = bnfinit(x^20 - 5*x^19 - 24*x^18 + 94*x^17 + 316*x^16 - 527*x^15 - 2449*x^14 + 241*x^13 + 8088*x^12 + 1094*x^11 - 4294*x^10 + 30738*x^9 + 13527*x^8 - 70159*x^7 - 81531*x^6 + 172477*x^5 + 311280*x^4 - 349102*x^3 - 259484*x^2 + 366509*x + 141791, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 24 x^{18} + 94 x^{17} + 316 x^{16} - 527 x^{15} - 2449 x^{14} + 241 x^{13} + 8088 x^{12} + 1094 x^{11} - 4294 x^{10} + 30738 x^{9} + 13527 x^{8} - 70159 x^{7} - 81531 x^{6} + 172477 x^{5} + 311280 x^{4} - 349102 x^{3} - 259484 x^{2} + 366509 x + 141791 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14728777159439278684044189453125=5^{13}\cdot 97^{2}\cdot 1039^{4}\cdot 1049^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.17$
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:  $5, 97, 1039, 1049$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{25607956099402057015806403441586559029780968944352335218951} a^{19} - \frac{1372470303944517474071957167069194548045729003056478733210}{25607956099402057015806403441586559029780968944352335218951} a^{18} + \frac{300488523507807515940660194080597686411493095425328147003}{25607956099402057015806403441586559029780968944352335218951} a^{17} - \frac{2405392001096758352683610921007672309727273846475866837734}{25607956099402057015806403441586559029780968944352335218951} a^{16} - \frac{1786129458778493650344839464013093691792706482857922522543}{25607956099402057015806403441586559029780968944352335218951} a^{15} + \frac{3543872712027002164846508217561025913437383236500849143512}{25607956099402057015806403441586559029780968944352335218951} a^{14} - \frac{11429934307281438018326734338290618052067872925435178977424}{25607956099402057015806403441586559029780968944352335218951} a^{13} + \frac{8753092812397681998572928895443537512218354289715713281825}{25607956099402057015806403441586559029780968944352335218951} a^{12} - \frac{3880303726898573425112335343311478698842125049116254912738}{25607956099402057015806403441586559029780968944352335218951} a^{11} - \frac{5542532335676819638696302739689457904730951060281751284885}{25607956099402057015806403441586559029780968944352335218951} a^{10} + \frac{791790884576223001649689977826208176332510293762855323477}{25607956099402057015806403441586559029780968944352335218951} a^{9} + \frac{4187738759395528853613195775240629684835123526973397602461}{25607956099402057015806403441586559029780968944352335218951} a^{8} - \frac{235968855092920586109667867003085928827170753106703458931}{25607956099402057015806403441586559029780968944352335218951} a^{7} - \frac{10560661596771684788778867834151501321530563365098825988505}{25607956099402057015806403441586559029780968944352335218951} a^{6} - \frac{11673394891232299133561353200270369812765865171075975458579}{25607956099402057015806403441586559029780968944352335218951} a^{5} - \frac{12442326983680772628272858268431940626946441149249564229828}{25607956099402057015806403441586559029780968944352335218951} a^{4} + \frac{11549294751552397098556976131264750903562439196065360330910}{25607956099402057015806403441586559029780968944352335218951} a^{3} - \frac{951854990576611316348403644476801290525438975374171586190}{1969842776877081308908184880122043002290843764950179632227} a^{2} + \frac{1209444392956178030655573798090991049221572114925596471330}{25607956099402057015806403441586559029780968944352335218951} a + \frac{768041689484536772140476643016309900371583027814589496395}{1969842776877081308908184880122043002290843764950179632227}$

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:  $13$
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:  \( 77014797.8746 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1039:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 378 conjugacy class representatives for t20n1039 are not computed
Character table for t20n1039 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.4.3405971875.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
1039Data not computed
1049Data not computed