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Properties

Label 20.8.49641192728...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{24}\cdot 5^{10}\cdot 13^{6}\cdot 29^{4}\cdot 31^{6}$
Root discriminant $60.92$
Ramified primes $2, 5, 13, 29, 31$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2848669, 5884406, -3561028, -4295009, 5591073, -1565642, -1187858, 846882, -145851, 6736, -30482, 18679, 4181, -5168, 1950, -182, -185, 74, -10, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 10*x^18 + 74*x^17 - 185*x^16 - 182*x^15 + 1950*x^14 - 5168*x^13 + 4181*x^12 + 18679*x^11 - 30482*x^10 + 6736*x^9 - 145851*x^8 + 846882*x^7 - 1187858*x^6 - 1565642*x^5 + 5591073*x^4 - 4295009*x^3 - 3561028*x^2 + 5884406*x - 2848669)
 
gp: K = bnfinit(x^20 - 3*x^19 - 10*x^18 + 74*x^17 - 185*x^16 - 182*x^15 + 1950*x^14 - 5168*x^13 + 4181*x^12 + 18679*x^11 - 30482*x^10 + 6736*x^9 - 145851*x^8 + 846882*x^7 - 1187858*x^6 - 1565642*x^5 + 5591073*x^4 - 4295009*x^3 - 3561028*x^2 + 5884406*x - 2848669, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 10 x^{18} + 74 x^{17} - 185 x^{16} - 182 x^{15} + 1950 x^{14} - 5168 x^{13} + 4181 x^{12} + 18679 x^{11} - 30482 x^{10} + 6736 x^{9} - 145851 x^{8} + 846882 x^{7} - 1187858 x^{6} - 1565642 x^{5} + 5591073 x^{4} - 4295009 x^{3} - 3561028 x^{2} + 5884406 x - 2848669 \)

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:  \(496411927280973952950108160000000000=2^{24}\cdot 5^{10}\cdot 13^{6}\cdot 29^{4}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.92$
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:  $2, 5, 13, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13} a^{16} + \frac{4}{13} a^{15} - \frac{2}{13} a^{14} - \frac{1}{13} a^{13} + \frac{1}{13} a^{12} - \frac{2}{13} a^{11} + \frac{1}{13} a^{10} - \frac{5}{13} a^{9} + \frac{5}{13} a^{8} - \frac{2}{13} a^{7} + \frac{1}{13} a^{6} + \frac{6}{13} a^{5} + \frac{1}{13} a^{4} - \frac{4}{13} a^{3} + \frac{4}{13} a^{2} + \frac{4}{13} a - \frac{5}{13}$, $\frac{1}{13} a^{17} - \frac{5}{13} a^{15} - \frac{6}{13} a^{14} + \frac{5}{13} a^{13} - \frac{6}{13} a^{12} - \frac{4}{13} a^{11} + \frac{4}{13} a^{10} - \frac{1}{13} a^{9} + \frac{4}{13} a^{8} - \frac{4}{13} a^{7} + \frac{2}{13} a^{6} + \frac{3}{13} a^{5} + \frac{5}{13} a^{4} - \frac{6}{13} a^{3} + \frac{1}{13} a^{2} + \frac{5}{13} a - \frac{6}{13}$, $\frac{1}{13} a^{18} + \frac{1}{13} a^{15} - \frac{5}{13} a^{14} + \frac{2}{13} a^{13} + \frac{1}{13} a^{12} - \frac{6}{13} a^{11} + \frac{4}{13} a^{10} + \frac{5}{13} a^{9} - \frac{5}{13} a^{8} + \frac{5}{13} a^{7} - \frac{5}{13} a^{6} - \frac{4}{13} a^{5} - \frac{1}{13} a^{4} - \frac{6}{13} a^{3} - \frac{1}{13} a^{2} + \frac{1}{13} a + \frac{1}{13}$, $\frac{1}{191479427222873960681950763998192611451474709575189524693575783} a^{19} + \frac{2919290353042582947243828523181291926286264236765723551257815}{191479427222873960681950763998192611451474709575189524693575783} a^{18} - \frac{33243328615518740965993863375847160433593752048439988142353}{11263495718992585922467691999893683026557335857364089687857399} a^{17} + \frac{3802407199216663461377936209077878040385917263147546331406241}{191479427222873960681950763998192611451474709575189524693575783} a^{16} + \frac{51947474106064610441708990632499581305110264139293443566051918}{191479427222873960681950763998192611451474709575189524693575783} a^{15} + \frac{60430089713274197848255562957515276944328541182650177177519370}{191479427222873960681950763998192611451474709575189524693575783} a^{14} + \frac{22473423771614222494012150981152937466401325327190196239997524}{191479427222873960681950763998192611451474709575189524693575783} a^{13} - \frac{850940837325281301730858279635910530252092867396895569348404}{14729186709451843129380827999860970111651900736553040361044291} a^{12} - \frac{85505377269567718043244600297445095180862287240400709128971302}{191479427222873960681950763998192611451474709575189524693575783} a^{11} + \frac{48446058606324467881821400677064002499951219869913373332189280}{191479427222873960681950763998192611451474709575189524693575783} a^{10} + \frac{220083070765430425761741832706968803516405249018126865460168}{191479427222873960681950763998192611451474709575189524693575783} a^{9} - \frac{15565548467440160836800297410359229885500124486616461175166}{866422747614814301728283999991821771273641219797237668296723} a^{8} - \frac{65453806113662954249439672353464402689307747837922097953787176}{191479427222873960681950763998192611451474709575189524693575783} a^{7} - \frac{24949693477801567914637459216086294464797430823637451467828244}{191479427222873960681950763998192611451474709575189524693575783} a^{6} - \frac{41400042104479807956583476084027209852679440617074231475099690}{191479427222873960681950763998192611451474709575189524693575783} a^{5} - \frac{80147775779830312386875750231625382921187121073726719010185184}{191479427222873960681950763998192611451474709575189524693575783} a^{4} + \frac{7190523423847805015068354012449343192493520631384989695220420}{191479427222873960681950763998192611451474709575189524693575783} a^{3} + \frac{4076059517185601647944365514256836781292317080784255459518116}{11263495718992585922467691999893683026557335857364089687857399} a^{2} - \frac{31604811864386038027538419759561465574560859513285316219211109}{191479427222873960681950763998192611451474709575189524693575783} a - \frac{29684595383980813287682681778861680742488064607944591522206281}{191479427222873960681950763998192611451474709575189524693575783}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 18088319570.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.109268775200000.1

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

Sibling fields

Degree 20 sibling: 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
5Data not computed
$13$13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.10.0.1$x^{10} + x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.6.5.3$x^{6} - 74431$$6$$1$$5$$C_6$$[\ ]_{6}$
31.8.0.1$x^{8} - x + 22$$1$$8$$0$$C_8$$[\ ]^{8}$