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Properties

Label 20.6.78574805586...9483.5
Degree $20$
Signature $[6, 7]$
Discriminant $-\,13^{10}\cdot 97^{2}\cdot 347^{7}$
Root discriminant $44.13$
Ramified primes $13, 97, 347$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T796

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-47837, -125280, -1855645, -864833, 303069, 1040532, 894721, -41335, -29854, 30308, 42877, 12828, -12271, 6146, -546, -71, -49, 19, 5, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 5*x^18 + 19*x^17 - 49*x^16 - 71*x^15 - 546*x^14 + 6146*x^13 - 12271*x^12 + 12828*x^11 + 42877*x^10 + 30308*x^9 - 29854*x^8 - 41335*x^7 + 894721*x^6 + 1040532*x^5 + 303069*x^4 - 864833*x^3 - 1855645*x^2 - 125280*x - 47837)
 
gp: K = bnfinit(x^20 - 6*x^19 + 5*x^18 + 19*x^17 - 49*x^16 - 71*x^15 - 546*x^14 + 6146*x^13 - 12271*x^12 + 12828*x^11 + 42877*x^10 + 30308*x^9 - 29854*x^8 - 41335*x^7 + 894721*x^6 + 1040532*x^5 + 303069*x^4 - 864833*x^3 - 1855645*x^2 - 125280*x - 47837, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 5 x^{18} + 19 x^{17} - 49 x^{16} - 71 x^{15} - 546 x^{14} + 6146 x^{13} - 12271 x^{12} + 12828 x^{11} + 42877 x^{10} + 30308 x^{9} - 29854 x^{8} - 41335 x^{7} + 894721 x^{6} + 1040532 x^{5} + 303069 x^{4} - 864833 x^{3} - 1855645 x^{2} - 125280 x - 47837 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-785748055861331850537130300409483=-\,13^{10}\cdot 97^{2}\cdot 347^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.13$
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:  $13, 97, 347$
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}{115420013686920817772442338952432957445138329568360925918888165534281} a^{19} + \frac{53088828997116613415826778614648593691652096048367374134015436502379}{115420013686920817772442338952432957445138329568360925918888165534281} a^{18} + \frac{9864656081974593561707406020857594496569584103442998298803267398176}{115420013686920817772442338952432957445138329568360925918888165534281} a^{17} + \frac{6111498910672505687024186550944588957228014624179675853320720001059}{115420013686920817772442338952432957445138329568360925918888165534281} a^{16} - \frac{2045014827066123801140894927222196142499310974956179640673689308304}{115420013686920817772442338952432957445138329568360925918888165534281} a^{15} - \frac{8992579704317153879293053432711230613046052409414328297499947663694}{115420013686920817772442338952432957445138329568360925918888165534281} a^{14} - \frac{39393436064526991316075758750408172197080630703755882663254912480268}{115420013686920817772442338952432957445138329568360925918888165534281} a^{13} - \frac{24748046135699114299090828013015207284298622052549627835061566350769}{115420013686920817772442338952432957445138329568360925918888165534281} a^{12} - \frac{6921981332803088202290638162132676282865605219734304235974690980265}{115420013686920817772442338952432957445138329568360925918888165534281} a^{11} - \frac{19210035043188742930700199881588105219153544114989163198760702453482}{115420013686920817772442338952432957445138329568360925918888165534281} a^{10} - \frac{50748548708892687804715066238909929070845628554493552291596310099009}{115420013686920817772442338952432957445138329568360925918888165534281} a^{9} - \frac{1813665496748397242264795986244192621970058286219119353460428897641}{115420013686920817772442338952432957445138329568360925918888165534281} a^{8} - \frac{38734727034453061440518358404880744621078326159888491144487619041026}{115420013686920817772442338952432957445138329568360925918888165534281} a^{7} + \frac{56159821638662791711018283229014175226116666699262964955159642106280}{115420013686920817772442338952432957445138329568360925918888165534281} a^{6} - \frac{22280826033858135579906974442303499370314155588467282658074215769362}{115420013686920817772442338952432957445138329568360925918888165534281} a^{5} + \frac{43709755360747257822051423449871867588872365579870062226019426329337}{115420013686920817772442338952432957445138329568360925918888165534281} a^{4} - \frac{2244227049104346577189230032819129964612170472335925899812867289687}{115420013686920817772442338952432957445138329568360925918888165534281} a^{3} - \frac{12779770174980450775353851948824615957859321131885398629100493201275}{115420013686920817772442338952432957445138329568360925918888165534281} a^{2} + \frac{54393834980457432252886399174182892230734782230959866531534493077088}{115420013686920817772442338952432957445138329568360925918888165534281} a - \frac{55218726830381553204055715352267564497977007246105659524178043552321}{115420013686920817772442338952432957445138329568360925918888165534281}$

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:  $12$
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:  \( 377351765.349 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T796:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n796 are not computed
Character table for t20n796 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.3.4511.1, 10.6.44707018837.1

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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
347Data not computed