Properties

Label 20.0.15913686444...8669.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 1609^{4}\cdot 4021$
Root discriminant $11.48$
Ramified primes $3, 1609, 4021$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois Group 20T887

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -2, 15, -1, 2, 6, -59, 17, 31, 42, -38, -42, 8, 50, -19, -13, 0, 11, -6, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 11*x^18 - 13*x^16 - 19*x^15 + 50*x^14 + 8*x^13 - 42*x^12 - 38*x^11 + 42*x^10 + 31*x^9 + 17*x^8 - 59*x^7 + 6*x^6 + 2*x^5 - x^4 + 15*x^3 - 2*x^2 - 3*x + 1)
gp: K = bnfinit(x^20 - 6*x^19 + 11*x^18 - 13*x^16 - 19*x^15 + 50*x^14 + 8*x^13 - 42*x^12 - 38*x^11 + 42*x^10 + 31*x^9 + 17*x^8 - 59*x^7 + 6*x^6 + 2*x^5 - x^4 + 15*x^3 - 2*x^2 - 3*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 6 x^{19} \) \(\mathstrut +\mathstrut 11 x^{18} \) \(\mathstrut -\mathstrut 13 x^{16} \) \(\mathstrut -\mathstrut 19 x^{15} \) \(\mathstrut +\mathstrut 50 x^{14} \) \(\mathstrut +\mathstrut 8 x^{13} \) \(\mathstrut -\mathstrut 42 x^{12} \) \(\mathstrut -\mathstrut 38 x^{11} \) \(\mathstrut +\mathstrut 42 x^{10} \) \(\mathstrut +\mathstrut 31 x^{9} \) \(\mathstrut +\mathstrut 17 x^{8} \) \(\mathstrut -\mathstrut 59 x^{7} \) \(\mathstrut +\mathstrut 6 x^{6} \) \(\mathstrut +\mathstrut 2 x^{5} \) \(\mathstrut -\mathstrut x^{4} \) \(\mathstrut +\mathstrut 15 x^{3} \) \(\mathstrut -\mathstrut 2 x^{2} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut +\mathstrut 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[0, 10]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(1591368644495819328669=3^{10}\cdot 1609^{4}\cdot 4021\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.48$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 1609, 4021$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{73} a^{18} - \frac{22}{73} a^{17} - \frac{23}{73} a^{16} + \frac{27}{73} a^{15} - \frac{35}{73} a^{14} - \frac{26}{73} a^{13} + \frac{33}{73} a^{12} + \frac{26}{73} a^{11} + \frac{17}{73} a^{10} + \frac{20}{73} a^{9} + \frac{22}{73} a^{8} - \frac{11}{73} a^{7} + \frac{23}{73} a^{6} + \frac{23}{73} a^{5} + \frac{31}{73} a^{4} - \frac{28}{73} a^{3} + \frac{15}{73} a^{2} - \frac{2}{73} a + \frac{7}{73}$, $\frac{1}{14026147} a^{19} + \frac{46492}{14026147} a^{18} + \frac{2911223}{14026147} a^{17} + \frac{2392303}{14026147} a^{16} - \frac{2529864}{14026147} a^{15} + \frac{6752822}{14026147} a^{14} + \frac{5119915}{14026147} a^{13} - \frac{1906743}{14026147} a^{12} - \frac{4495788}{14026147} a^{11} + \frac{2465816}{14026147} a^{10} - \frac{2667284}{14026147} a^{9} + \frac{4850906}{14026147} a^{8} + \frac{267352}{14026147} a^{7} + \frac{5511968}{14026147} a^{6} - \frac{5833173}{14026147} a^{5} + \frac{5215422}{14026147} a^{4} + \frac{2296085}{14026147} a^{3} + \frac{6128470}{14026147} a^{2} - \frac{4364178}{14026147} a + \frac{2440481}{14026147}$

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

Class group and class number

Trivial Abelian group, 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:  $9$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( -\frac{82}{73} a^{19} + \frac{435}{73} a^{18} - \frac{627}{73} a^{17} - 4 a^{16} + \frac{655}{73} a^{15} + \frac{1940}{73} a^{14} - \frac{2590}{73} a^{13} - \frac{1830}{73} a^{12} + \frac{1410}{73} a^{11} + \frac{3630}{73} a^{10} - \frac{714}{73} a^{9} - \frac{2060}{73} a^{8} - \frac{2887}{73} a^{7} + \frac{2616}{73} a^{6} + \frac{354}{73} a^{5} + \frac{591}{73} a^{4} + \frac{237}{73} a^{3} - \frac{661}{73} a^{2} - \frac{99}{73} a + \frac{126}{73} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{6091276}{14026147} a^{19} - \frac{25159907}{14026147} a^{18} + \frac{8997927}{14026147} a^{17} + \frac{75373549}{14026147} a^{16} - \frac{24452770}{14026147} a^{15} - \frac{197087877}{14026147} a^{14} + \frac{26815742}{14026147} a^{13} + \frac{353863516}{14026147} a^{12} + \frac{42091283}{14026147} a^{11} - \frac{380032631}{14026147} a^{10} - \frac{240020198}{14026147} a^{9} + \frac{213591143}{14026147} a^{8} + \frac{356411029}{14026147} a^{7} + \frac{41131374}{14026147} a^{6} - \frac{218996892}{14026147} a^{5} - \frac{39592315}{14026147} a^{4} - \frac{76764613}{14026147} a^{3} + \frac{2797946}{14026147} a^{2} + \frac{44729688}{14026147} a - \frac{2178519}{14026147} \),  \( \frac{15993462}{14026147} a^{19} - \frac{86558157}{14026147} a^{18} + \frac{1793881}{192139} a^{17} + \frac{49085547}{14026147} a^{16} - \frac{147803598}{14026147} a^{15} - \frac{361361032}{14026147} a^{14} + \frac{565730951}{14026147} a^{13} + \frac{319319854}{14026147} a^{12} - \frac{377890415}{14026147} a^{11} - \frac{688718057}{14026147} a^{10} + \frac{266624319}{14026147} a^{9} + \frac{436144362}{14026147} a^{8} + \frac{487338726}{14026147} a^{7} - \frac{605283641}{14026147} a^{6} - \frac{53236751}{14026147} a^{5} - \frac{72544924}{14026147} a^{4} - \frac{45830047}{14026147} a^{3} + \frac{169248534}{14026147} a^{2} + \frac{32975494}{14026147} a - \frac{27483849}{14026147} \),  \( \frac{10117353}{14026147} a^{19} - \frac{59841870}{14026147} a^{18} + \frac{100860572}{14026147} a^{17} + \frac{31789519}{14026147} a^{16} - \frac{148147317}{14026147} a^{15} - \frac{241577026}{14026147} a^{14} + \frac{498866903}{14026147} a^{13} + \frac{254328426}{14026147} a^{12} - \frac{458887443}{14026147} a^{11} - \frac{588851910}{14026147} a^{10} + \frac{349870105}{14026147} a^{9} + \frac{542779785}{14026147} a^{8} + \frac{318135101}{14026147} a^{7} - \frac{636538029}{14026147} a^{6} - \frac{192398509}{14026147} a^{5} + \frac{20298702}{14026147} a^{4} + \frac{25260783}{14026147} a^{3} + \frac{178674866}{14026147} a^{2} + \frac{42441970}{14026147} a - \frac{34636200}{14026147} \),  \( \frac{2390396}{14026147} a^{19} - \frac{3902382}{14026147} a^{18} - \frac{24583339}{14026147} a^{17} + \frac{60647848}{14026147} a^{16} + \frac{19165308}{14026147} a^{15} - \frac{93136311}{14026147} a^{14} - \frac{132323650}{14026147} a^{13} + \frac{230744154}{14026147} a^{12} + \frac{151882287}{14026147} a^{11} - \frac{112622029}{14026147} a^{10} - \frac{292182510}{14026147} a^{9} - \frac{45556}{14026147} a^{8} + \frac{145229935}{14026147} a^{7} + \frac{243607942}{14026147} a^{6} - \frac{99800833}{14026147} a^{5} + \frac{22147078}{14026147} a^{4} - \frac{116981661}{14026147} a^{3} - \frac{64125820}{14026147} a^{2} + \frac{268104}{192139} a - \frac{4070466}{14026147} \),  \( \frac{7480162}{14026147} a^{19} - \frac{31950590}{14026147} a^{18} + \frac{19264687}{14026147} a^{17} + \frac{72926297}{14026147} a^{16} - \frac{24369358}{14026147} a^{15} - \frac{213704300}{14026147} a^{14} + \frac{51938913}{14026147} a^{13} + \frac{4488287}{192139} a^{12} + \frac{56362181}{14026147} a^{11} - \frac{351304335}{14026147} a^{10} - \frac{229384028}{14026147} a^{9} + \frac{124489390}{14026147} a^{8} + \frac{363094953}{14026147} a^{7} + \frac{63941449}{14026147} a^{6} - \frac{133783885}{14026147} a^{5} - \frac{69624238}{14026147} a^{4} - \frac{103505803}{14026147} a^{3} + \frac{312899}{14026147} a^{2} + \frac{47321056}{14026147} a - \frac{400728}{14026147} \),  \( \frac{9878730}{14026147} a^{19} - \frac{50381715}{14026147} a^{18} + \frac{62946208}{14026147} a^{17} + \frac{58588940}{14026147} a^{16} - \frac{79454051}{14026147} a^{15} - \frac{255143887}{14026147} a^{14} + \frac{259073755}{14026147} a^{13} + \frac{320129432}{14026147} a^{12} - \frac{136365688}{14026147} a^{11} - \frac{484222371}{14026147} a^{10} - \frac{37596529}{14026147} a^{9} + \frac{286478780}{14026147} a^{8} + \frac{411859085}{14026147} a^{7} - \frac{203267901}{14026147} a^{6} - \frac{140938271}{14026147} a^{5} - \frac{64037649}{14026147} a^{4} - \frac{91289620}{14026147} a^{3} + \frac{87074128}{14026147} a^{2} + \frac{18675767}{14026147} a - \frac{4120106}{14026147} \),  \( \frac{955036}{14026147} a^{19} - \frac{9474748}{14026147} a^{18} + \frac{26670641}{14026147} a^{17} - \frac{12609911}{14026147} a^{16} - \frac{39165009}{14026147} a^{15} - \frac{12567301}{14026147} a^{14} + \frac{143913631}{14026147} a^{13} - \frac{20826476}{14026147} a^{12} - \frac{163355265}{14026147} a^{11} - \frac{76813469}{14026147} a^{10} + \frac{188400814}{14026147} a^{9} + \frac{127010033}{14026147} a^{8} - \frac{24828413}{14026147} a^{7} - \frac{220765085}{14026147} a^{6} + \frac{1790780}{14026147} a^{5} + \frac{29795959}{14026147} a^{4} + \frac{44238969}{14026147} a^{3} + \frac{43501184}{14026147} a^{2} - \frac{12960654}{14026147} a - \frac{98960}{192139} \),  \( \frac{21634329}{14026147} a^{19} - \frac{104755234}{14026147} a^{18} + \frac{120005737}{14026147} a^{17} + \frac{120749972}{14026147} a^{16} - \frac{109911452}{14026147} a^{15} - \frac{545830644}{14026147} a^{14} + \frac{438776760}{14026147} a^{13} + \frac{623179172}{14026147} a^{12} - \frac{84225939}{14026147} a^{11} - \frac{940255405}{14026147} a^{10} - \frac{2493635}{192139} a^{9} + \frac{399772380}{14026147} a^{8} + \frac{868911354}{14026147} a^{7} - \frac{318085502}{14026147} a^{6} - \frac{134533837}{14026147} a^{5} - \frac{198062314}{14026147} a^{4} - \frac{147321640}{14026147} a^{3} + \frac{70422303}{14026147} a^{2} + \frac{35701495}{14026147} a - \frac{12823039}{14026147} \),  \( \frac{6804340}{14026147} a^{19} - \frac{37732653}{14026147} a^{18} + \frac{59069995}{14026147} a^{17} + \frac{20382294}{14026147} a^{16} - \frac{70674087}{14026147} a^{15} - \frac{160833164}{14026147} a^{14} + \frac{270142724}{14026147} a^{13} + \frac{147078598}{14026147} a^{12} - \frac{194358734}{14026147} a^{11} - \frac{341916758}{14026147} a^{10} + \frac{149627745}{14026147} a^{9} + \frac{242483102}{14026147} a^{8} + \frac{227601461}{14026147} a^{7} - \frac{312423623}{14026147} a^{6} - \frac{60505872}{14026147} a^{5} - \frac{36735629}{14026147} a^{4} + \frac{5409963}{14026147} a^{3} + \frac{78093580}{14026147} a^{2} + \frac{30677903}{14026147} a - \frac{24769813}{14026147} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 427.42684139 \) (assuming GRH)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

20T887:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 245760
The 201 conjugacy class representatives for t20n887 are not computed
Character table for t20n887 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.1.1609.1, 10.0.629098083.1

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 $20$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
3Data not computed
1609Data not computed
4021Data not computed