Properties

Label 20.0.17235683651...5632.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{15}\cdot 47^{10}$
Root discriminant $11.53$
Ramified primes $2, 47$
Class number $1$
Class group Trivial
Galois Group $C_5:D_4$ (as 20T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 21, -35, 37, -34, 26, -3, 17, -56, 29, 19, 2, -28, 3, 15, -3, -4, 1, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + x^18 - 4*x^17 - 3*x^16 + 15*x^15 + 3*x^14 - 28*x^13 + 2*x^12 + 19*x^11 + 29*x^10 - 56*x^9 + 17*x^8 - 3*x^7 + 26*x^6 - 34*x^5 + 37*x^4 - 35*x^3 + 21*x^2 - 7*x + 1)
gp: K = bnfinit(x^20 - x^19 + x^18 - 4*x^17 - 3*x^16 + 15*x^15 + 3*x^14 - 28*x^13 + 2*x^12 + 19*x^11 + 29*x^10 - 56*x^9 + 17*x^8 - 3*x^7 + 26*x^6 - 34*x^5 + 37*x^4 - 35*x^3 + 21*x^2 - 7*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut x^{19} \) \(\mathstrut +\mathstrut x^{18} \) \(\mathstrut -\mathstrut 4 x^{17} \) \(\mathstrut -\mathstrut 3 x^{16} \) \(\mathstrut +\mathstrut 15 x^{15} \) \(\mathstrut +\mathstrut 3 x^{14} \) \(\mathstrut -\mathstrut 28 x^{13} \) \(\mathstrut +\mathstrut 2 x^{12} \) \(\mathstrut +\mathstrut 19 x^{11} \) \(\mathstrut +\mathstrut 29 x^{10} \) \(\mathstrut -\mathstrut 56 x^{9} \) \(\mathstrut +\mathstrut 17 x^{8} \) \(\mathstrut -\mathstrut 3 x^{7} \) \(\mathstrut +\mathstrut 26 x^{6} \) \(\mathstrut -\mathstrut 34 x^{5} \) \(\mathstrut +\mathstrut 37 x^{4} \) \(\mathstrut -\mathstrut 35 x^{3} \) \(\mathstrut +\mathstrut 21 x^{2} \) \(\mathstrut -\mathstrut 7 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(1723568365103679045632=2^{15}\cdot 47^{10}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.53$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{47} a^{18} - \frac{16}{47} a^{17} - \frac{15}{47} a^{16} - \frac{7}{47} a^{15} - \frac{6}{47} a^{14} + \frac{17}{47} a^{13} + \frac{15}{47} a^{12} + \frac{1}{47} a^{11} + \frac{1}{47} a^{10} - \frac{17}{47} a^{9} - \frac{19}{47} a^{8} + \frac{22}{47} a^{7} - \frac{8}{47} a^{6} - \frac{16}{47} a^{5} + \frac{11}{47} a^{4} - \frac{4}{47} a^{3} + \frac{7}{47} a^{2} - \frac{9}{47} a + \frac{9}{47}$, $\frac{1}{24657195037} a^{19} + \frac{14423459}{24657195037} a^{18} + \frac{6052526827}{24657195037} a^{17} + \frac{6076674939}{24657195037} a^{16} + \frac{490195653}{24657195037} a^{15} - \frac{3702211118}{24657195037} a^{14} - \frac{3221954579}{24657195037} a^{13} - \frac{1683844526}{24657195037} a^{12} - \frac{5510042910}{24657195037} a^{11} + \frac{6610333898}{24657195037} a^{10} - \frac{215387826}{524621171} a^{9} + \frac{2552398449}{24657195037} a^{8} - \frac{8641400464}{24657195037} a^{7} + \frac{5276938446}{24657195037} a^{6} + \frac{7916454968}{24657195037} a^{5} - \frac{7803064182}{24657195037} a^{4} - \frac{4935834703}{24657195037} a^{3} + \frac{6673552933}{24657195037} a^{2} - \frac{4891694897}{24657195037} a - \frac{6491582894}{24657195037}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{5480998887964}{24657195037} a^{19} - \frac{2705858205680}{24657195037} a^{18} + \frac{4109065027076}{24657195037} a^{17} - \frac{19843517699273}{24657195037} a^{16} - \frac{26491863446166}{24657195037} a^{15} + \frac{68807124820422}{24657195037} a^{14} + \frac{51292712573652}{24657195037} a^{13} - \frac{127516497779395}{24657195037} a^{12} - \frac{53628022468519}{24657195037} a^{11} + \frac{77019068384271}{24657195037} a^{10} + \frac{197977878613986}{24657195037} a^{9} - \frac{206701440792921}{24657195037} a^{8} - \frac{11550650649486}{24657195037} a^{7} - \frac{22268111751795}{24657195037} a^{6} + \frac{131256349241091}{24657195037} a^{5} - \frac{119856650893652}{24657195037} a^{4} + \frac{142071308290184}{24657195037} a^{3} - \frac{119890461913823}{24657195037} a^{2} + \frac{54344702035321}{24657195037} a - \frac{10799261699363}{24657195037} \),  \( \frac{2583028267356}{24657195037} a^{19} - \frac{1285060972434}{24657195037} a^{18} + \frac{1941863590891}{24657195037} a^{17} - \frac{9357580953053}{24657195037} a^{16} - \frac{12449574917915}{24657195037} a^{15} + \frac{32472166085848}{24657195037} a^{14} + \frac{24039487028713}{24657195037} a^{13} - \frac{60189883168722}{24657195037} a^{12} - \frac{25011563347548}{24657195037} a^{11} + \frac{36410535757503}{24657195037} a^{10} + \frac{93103773144411}{24657195037} a^{9} - \frac{97807299591623}{24657195037} a^{8} - \frac{5009784095596}{24657195037} a^{7} - \frac{10388115733700}{24657195037} a^{6} + \frac{61838894141826}{24657195037} a^{5} - \frac{56793053157113}{24657195037} a^{4} + \frac{67177484198107}{24657195037} a^{3} - \frac{56700382590493}{24657195037} a^{2} + \frac{25819632846469}{24657195037} a - \frac{5174012966636}{24657195037} \),  \( \frac{5436226585702}{24657195037} a^{19} - \frac{2677155629433}{24657195037} a^{18} + \frac{4081228867994}{24657195037} a^{17} - \frac{19675628703414}{24657195037} a^{16} - \frac{26293148498288}{24657195037} a^{15} + \frac{68184904889870}{24657195037} a^{14} + \frac{50896976895063}{24657195037} a^{13} - \frac{126330380545760}{24657195037} a^{12} - \frac{53202467360930}{24657195037} a^{11} + \frac{76183846204613}{24657195037} a^{10} + \frac{196266236199580}{24657195037} a^{9} - \frac{204733971595442}{24657195037} a^{8} - \frac{11340696967805}{24657195037} a^{7} - \frac{22232533113166}{24657195037} a^{6} + \frac{2766420392906}{524621171} a^{5} - \frac{118824086068186}{24657195037} a^{4} + \frac{140935586048729}{24657195037} a^{3} - \frac{118840635217277}{24657195037} a^{2} + \frac{53932456382740}{24657195037} a - \frac{10757472487966}{24657195037} \),  \( \frac{2891478538813}{24657195037} a^{19} - \frac{1431792286119}{24657195037} a^{18} + \frac{2166692681531}{24657195037} a^{17} - \frac{10471684009602}{24657195037} a^{16} - \frac{13961336863929}{24657195037} a^{15} + \frac{36332500363225}{24657195037} a^{14} + \frac{27028894982022}{24657195037} a^{13} - \frac{67339901424706}{24657195037} a^{12} - \frac{28245538037761}{24657195037} a^{11} + \frac{40713409229703}{24657195037} a^{10} + \frac{104457666092036}{24657195037} a^{9} - \frac{109194684134552}{24657195037} a^{8} - \frac{6075315320426}{24657195037} a^{7} - \frac{11725614867254}{24657195037} a^{6} + \frac{69304548799725}{24657195037} a^{5} - \frac{63255405449180}{24657195037} a^{4} + \frac{74997506422845}{24657195037} a^{3} - \frac{63360919489344}{24657195037} a^{2} + \frac{28699883408514}{24657195037} a - \frac{5710585904942}{24657195037} \),  \( \frac{4066295860717}{24657195037} a^{19} - \frac{2016131149983}{24657195037} a^{18} + \frac{3049820185617}{24657195037} a^{17} - \frac{14727012290985}{24657195037} a^{16} - \frac{19624066735353}{24657195037} a^{15} + \frac{51101479050551}{24657195037} a^{14} + \frac{37962757928700}{24657195037} a^{13} - \frac{94718514287515}{24657195037} a^{12} - \frac{39619762805172}{24657195037} a^{11} + \frac{57280375743121}{24657195037} a^{10} + \frac{146795934632763}{24657195037} a^{9} - \frac{153679566070242}{24657195037} a^{8} - \frac{8352033506680}{24657195037} a^{7} - \frac{16431432182804}{24657195037} a^{6} + \frac{97418698299275}{24657195037} a^{5} - \frac{89104488350200}{24657195037} a^{4} + \frac{105546106930088}{24657195037} a^{3} - \frac{89129142439035}{24657195037} a^{2} + \frac{40423000155958}{24657195037} a - \frac{8051872106684}{24657195037} \),  \( \frac{1493295134129}{24657195037} a^{19} - \frac{734900395412}{24657195037} a^{18} + \frac{1123850845761}{24657195037} a^{17} - \frac{5401173454041}{24657195037} a^{16} - \frac{7220439895540}{24657195037} a^{15} + \frac{18719821541222}{24657195037} a^{14} + \frac{13957268039633}{24657195037} a^{13} - \frac{34697715676024}{24657195037} a^{12} - \frac{14567069894634}{24657195037} a^{11} + \frac{20934787988544}{24657195037} a^{10} + \frac{53843134167435}{24657195037} a^{9} - \frac{56293373014406}{24657195037} a^{8} - \frac{3041510187600}{24657195037} a^{7} - \frac{6026561138656}{24657195037} a^{6} + \frac{35696090048280}{24657195037} a^{5} - \frac{32718077590466}{24657195037} a^{4} + \frac{38701404352874}{24657195037} a^{3} - \frac{32631778279372}{24657195037} a^{2} + \frac{14851066977162}{24657195037} a - \frac{62829391209}{524621171} \),  \( \frac{4516761286788}{24657195037} a^{19} - \frac{2217128251905}{24657195037} a^{18} + \frac{3383102998920}{24657195037} a^{17} - \frac{16343242652908}{24657195037} a^{16} - \frac{21873120117310}{24657195037} a^{15} + \frac{56631907510053}{24657195037} a^{14} + \frac{42411936554957}{24657195037} a^{13} - \frac{104935298609050}{24657195037} a^{12} - \frac{44458739922892}{24657195037} a^{11} + \frac{63295804832621}{24657195037} a^{10} + \frac{163296706684474}{24657195037} a^{9} - \frac{169870529613723}{24657195037} a^{8} - \frac{9903036374799}{24657195037} a^{7} - \frac{18449706648143}{24657195037} a^{6} + \frac{108123268274122}{24657195037} a^{5} - \frac{98511846110402}{24657195037} a^{4} + \frac{116862389734695}{24657195037} a^{3} - \frac{98518002167963}{24657195037} a^{2} + \frac{44573775790867}{24657195037} a - \frac{8839759954053}{24657195037} \),  \( a \),  \( \frac{3859277408905}{24657195037} a^{19} - \frac{1925809512460}{24657195037} a^{18} + \frac{2895671005862}{24657195037} a^{17} - \frac{13985128294338}{24657195037} a^{16} - \frac{18583057361569}{24657195037} a^{15} + \frac{1033528773504}{524621171} a^{14} + \frac{35901438316249}{24657195037} a^{13} - \frac{90070128654302}{24657195037} a^{12} - \frac{37377150910160}{24657195037} a^{11} + \frac{54608577802149}{24657195037} a^{10} + \frac{139240653503031}{24657195037} a^{9} - \frac{146405809422505}{24657195037} a^{8} - \frac{7691561721632}{24657195037} a^{7} - \frac{15359598052206}{24657195037} a^{6} + \frac{92616109379485}{24657195037} a^{5} - \frac{84884784894300}{24657195037} a^{4} + \frac{100266453700579}{24657195037} a^{3} - \frac{84802150286496}{24657195037} a^{2} + \frac{38573598812824}{24657195037} a - \frac{7687157596026}{24657195037} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 143.606119616 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_5:D_4$ (as 20T11):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 40
The 13 conjugacy class representatives for $C_5:D_4$
Character table for $C_5:D_4$

Intermediate fields

\(\Q(\sqrt{-47}) \), 4.0.17672.1, 5.1.2209.1 x5, 10.0.229345007.1

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

Sibling fields

Galois closure: data not computed
Degree 20 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$