Properties

Label 20.0.28255841683...8373.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 37^{5}\cdot 109^{2}\cdot 241^{2}$
Root discriminant $11.82$
Ramified primes $3, 37, 109, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois Group 20T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 53, -190, 509, -1057, 1739, -2282, 2367, -1874, 1033, -274, -109, 140, -28, -58, 75, -52, 24, -7, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 24*x^18 - 52*x^17 + 75*x^16 - 58*x^15 - 28*x^14 + 140*x^13 - 109*x^12 - 274*x^11 + 1033*x^10 - 1874*x^9 + 2367*x^8 - 2282*x^7 + 1739*x^6 - 1057*x^5 + 509*x^4 - 190*x^3 + 53*x^2 - 10*x + 1)
gp: K = bnfinit(x^20 - 7*x^19 + 24*x^18 - 52*x^17 + 75*x^16 - 58*x^15 - 28*x^14 + 140*x^13 - 109*x^12 - 274*x^11 + 1033*x^10 - 1874*x^9 + 2367*x^8 - 2282*x^7 + 1739*x^6 - 1057*x^5 + 509*x^4 - 190*x^3 + 53*x^2 - 10*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 7 x^{19} \) \(\mathstrut +\mathstrut 24 x^{18} \) \(\mathstrut -\mathstrut 52 x^{17} \) \(\mathstrut +\mathstrut 75 x^{16} \) \(\mathstrut -\mathstrut 58 x^{15} \) \(\mathstrut -\mathstrut 28 x^{14} \) \(\mathstrut +\mathstrut 140 x^{13} \) \(\mathstrut -\mathstrut 109 x^{12} \) \(\mathstrut -\mathstrut 274 x^{11} \) \(\mathstrut +\mathstrut 1033 x^{10} \) \(\mathstrut -\mathstrut 1874 x^{9} \) \(\mathstrut +\mathstrut 2367 x^{8} \) \(\mathstrut -\mathstrut 2282 x^{7} \) \(\mathstrut +\mathstrut 1739 x^{6} \) \(\mathstrut -\mathstrut 1057 x^{5} \) \(\mathstrut +\mathstrut 509 x^{4} \) \(\mathstrut -\mathstrut 190 x^{3} \) \(\mathstrut +\mathstrut 53 x^{2} \) \(\mathstrut -\mathstrut 10 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(2825584168318748978373=3^{10}\cdot 37^{5}\cdot 109^{2}\cdot 241^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.82$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 37, 109, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{58974943} a^{19} + \frac{26100956}{58974943} a^{18} + \frac{4441811}{58974943} a^{17} + \frac{6717049}{58974943} a^{16} + \frac{29168546}{58974943} a^{15} + \frac{21041664}{58974943} a^{14} - \frac{1661676}{58974943} a^{13} + \frac{8787212}{58974943} a^{12} + \frac{21434130}{58974943} a^{11} + \frac{27183508}{58974943} a^{10} - \frac{7725163}{58974943} a^{9} + \frac{15959240}{58974943} a^{8} + \frac{10227602}{58974943} a^{7} + \frac{4964115}{58974943} a^{6} - \frac{3726288}{58974943} a^{5} + \frac{1530909}{58974943} a^{4} + \frac{21410941}{58974943} a^{3} - \frac{27182349}{58974943} a^{2} + \frac{4596607}{58974943} a + \frac{17015652}{58974943}$

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:  \( -\frac{603522}{140083} a^{19} + \frac{4064809}{140083} a^{18} - \frac{13277554}{140083} a^{17} + \frac{27061215}{140083} a^{16} - \frac{35690210}{140083} a^{15} + \frac{21127454}{140083} a^{14} + \frac{27499899}{140083} a^{13} - \frac{78545176}{140083} a^{12} + \frac{37794123}{140083} a^{11} + \frac{188496067}{140083} a^{10} - \frac{573679398}{140083} a^{9} + \frac{936961751}{140083} a^{8} - \frac{1079862509}{140083} a^{7} + \frac{948908718}{140083} a^{6} - \frac{654085764}{140083} a^{5} + \frac{354326695}{140083} a^{4} - \frac{148436724}{140083} a^{3} + \frac{46655711}{140083} a^{2} - \frac{10373793}{140083} a + \frac{1347044}{140083} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{170469271}{58974943} a^{19} - \frac{1107741039}{58974943} a^{18} + \frac{3494946958}{58974943} a^{17} - \frac{6929036835}{58974943} a^{16} + \frac{8932019192}{58974943} a^{15} - \frac{5006475513}{58974943} a^{14} - \frac{7278454937}{58974943} a^{13} + \frac{19215480418}{58974943} a^{12} - \frac{7094073266}{58974943} a^{11} - \frac{50888046390}{58974943} a^{10} + \frac{146458214592}{58974943} a^{9} - \frac{236071278443}{58974943} a^{8} + \frac{273396584995}{58974943} a^{7} - \frac{245014812337}{58974943} a^{6} + \frac{174367714885}{58974943} a^{5} - \frac{98869823579}{58974943} a^{4} + \frac{44056731443}{58974943} a^{3} - \frac{14979138902}{58974943} a^{2} + \frac{3589774868}{58974943} a - \frac{509835363}{58974943} \),  \( \frac{154535783}{58974943} a^{19} - \frac{991858053}{58974943} a^{18} + \frac{3127902055}{58974943} a^{17} - \frac{6165716435}{58974943} a^{16} + \frac{7772183881}{58974943} a^{15} - \frac{3861535384}{58974943} a^{14} - \frac{7489634184}{58974943} a^{13} + \frac{18095803770}{58974943} a^{12} - \frac{6129253312}{58974943} a^{11} - \frac{47779338876}{58974943} a^{10} + \frac{134020851972}{58974943} a^{9} - \frac{209619553989}{58974943} a^{8} + \frac{232367009724}{58974943} a^{7} - \frac{195423465312}{58974943} a^{6} + \frac{127400827323}{58974943} a^{5} - \frac{63637839084}{58974943} a^{4} + \frac{23381382653}{58974943} a^{3} - \frac{5828813705}{58974943} a^{2} + \frac{814615000}{58974943} a - \frac{1654540}{58974943} \),  \( \frac{1999557}{830633} a^{19} - \frac{11621030}{830633} a^{18} + \frac{32845725}{830633} a^{17} - \frac{57103703}{830633} a^{16} + \frac{59853676}{830633} a^{15} - \frac{6105135}{830633} a^{14} - \frac{103812724}{830633} a^{13} + \frac{160553047}{830633} a^{12} + \frac{43900364}{830633} a^{11} - \frac{605708363}{830633} a^{10} + \frac{1316763499}{830633} a^{9} - \frac{1770622082}{830633} a^{8} + \frac{1733831181}{830633} a^{7} - \frac{1302660169}{830633} a^{6} + \frac{761468086}{830633} a^{5} - \frac{339192181}{830633} a^{4} + \frac{113541589}{830633} a^{3} - \frac{27213490}{830633} a^{2} + \frac{5101620}{830633} a + \frac{94914}{830633} \),  \( \frac{207403991}{58974943} a^{19} - \frac{1329825328}{58974943} a^{18} + \frac{4221012307}{58974943} a^{17} - \frac{8449034836}{58974943} a^{16} + \frac{10991313885}{58974943} a^{15} - \frac{6242257941}{58974943} a^{14} - \frac{8836687422}{58974943} a^{13} + \frac{23941936121}{58974943} a^{12} - \frac{9987499377}{58974943} a^{11} - \frac{60740921145}{58974943} a^{10} + \frac{179482526555}{58974943} a^{9} - \frac{291225226680}{58974943} a^{8} + \frac{335849394540}{58974943} a^{7} - \frac{295797532191}{58974943} a^{6} + \frac{204216771182}{58974943} a^{5} - \frac{110375651809}{58974943} a^{4} + \frac{45895175881}{58974943} a^{3} - \frac{13980973735}{58974943} a^{2} + \frac{2921509570}{58974943} a - \frac{298975827}{58974943} \),  \( \frac{339008175}{58974943} a^{19} - \frac{2222710082}{58974943} a^{18} + \frac{7149431935}{58974943} a^{17} - \frac{14468632217}{58974943} a^{16} + \frac{19076855397}{58974943} a^{15} - \frac{11395818298}{58974943} a^{14} - \frac{14205694894}{58974943} a^{13} + \frac{40795922241}{58974943} a^{12} - \frac{18825932913}{58974943} a^{11} - \frac{100608348246}{58974943} a^{10} + \frac{304589105739}{58974943} a^{9} - \frac{500427996152}{58974943} a^{8} + \frac{584022637199}{58974943} a^{7} - \frac{522427035231}{58974943} a^{6} + \frac{368421625661}{58974943} a^{5} - \frac{205329234678}{58974943} a^{4} + \frac{89138569659}{58974943} a^{3} - \frac{29117121795}{58974943} a^{2} + \frac{6703309431}{58974943} a - \frac{847496673}{58974943} \),  \( \frac{84930488}{58974943} a^{19} - \frac{565770714}{58974943} a^{18} + \frac{1868219015}{58974943} a^{17} - \frac{3900003283}{58974943} a^{16} + \frac{5351989753}{58974943} a^{15} - \frac{3579886111}{58974943} a^{14} - \frac{3217575696}{58974943} a^{13} + \frac{11041043945}{58974943} a^{12} - \frac{6719563567}{58974943} a^{11} - \frac{24245314709}{58974943} a^{10} + \frac{80775696951}{58974943} a^{9} - \frac{138323796427}{58974943} a^{8} + \frac{165795191051}{58974943} a^{7} - \frac{150585130256}{58974943} a^{6} + \frac{106576826067}{58974943} a^{5} - \frac{58649972990}{58974943} a^{4} + \frac{24493316533}{58974943} a^{3} - \frac{7359999014}{58974943} a^{2} + \frac{1425021530}{58974943} a - \frac{88348972}{58974943} \),  \( \frac{285722443}{58974943} a^{19} - \frac{1916478553}{58974943} a^{18} + \frac{6314558012}{58974943} a^{17} - \frac{13104665215}{58974943} a^{16} + \frac{17856502623}{58974943} a^{15} - \frac{11807329174}{58974943} a^{14} - \frac{10963062708}{58974943} a^{13} + \frac{36737314065}{58974943} a^{12} - \frac{21212049505}{58974943} a^{11} - \frac{83203884151}{58974943} a^{10} + \frac{271082718441}{58974943} a^{9} - \frac{460894269782}{58974943} a^{8} + \frac{552183183424}{58974943} a^{7} - \frac{505170417507}{58974943} a^{6} + \frac{363433862005}{58974943} a^{5} - \frac{206028093207}{58974943} a^{4} + \frac{90762582803}{58974943} a^{3} - \frac{30019999917}{58974943} a^{2} + \frac{7042216505}{58974943} a - \frac{930602197}{58974943} \),  \( \frac{127459714}{58974943} a^{19} - \frac{614850438}{58974943} a^{18} + \frac{1294168561}{58974943} a^{17} - \frac{1200652422}{58974943} a^{16} - \frac{788930412}{58974943} a^{15} + \frac{5046995517}{58974943} a^{14} - \frac{8586291499}{58974943} a^{13} + \frac{3454907665}{58974943} a^{12} + \frac{16425072666}{58974943} a^{11} - \frac{40500540702}{58974943} a^{10} + \frac{43225776469}{58974943} a^{9} - \frac{10055142370}{58974943} a^{8} - \frac{40730282560}{58974943} a^{7} + \frac{75813075936}{58974943} a^{6} - \frac{78166782586}{58974943} a^{5} + \frac{56212015465}{58974943} a^{4} - \frac{29119672945}{58974943} a^{3} + \frac{10652795855}{58974943} a^{2} - \frac{2647369755}{58974943} a + \frac{357274570}{58974943} \),  \( \frac{229770728}{58974943} a^{19} - \frac{1544081300}{58974943} a^{18} + \frac{5088949017}{58974943} a^{17} - \frac{10594101342}{58974943} a^{16} + \frac{14550747408}{58974943} a^{15} - \frac{9907451157}{58974943} a^{14} - \frac{8272030832}{58974943} a^{13} + \frac{29199007803}{58974943} a^{12} - \frac{17486626224}{58974943} a^{11} - \frac{65564772528}{58974943} a^{10} + \frac{217094636634}{58974943} a^{9} - \frac{373475619867}{58974943} a^{8} + \frac{453387203084}{58974943} a^{7} - \frac{421305205266}{58974943} a^{6} + \frac{308267684496}{58974943} a^{5} - \frac{177963617784}{58974943} a^{4} + \frac{79700868526}{58974943} a^{3} - \frac{26580657274}{58974943} a^{2} + \frac{6010961122}{58974943} a - \frac{690618265}{58974943} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 567.746569822 \) (assuming GRH)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

20T781:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 115200
The 119 conjugacy class representatives for t20n781 are not computed
Character table for t20n781 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.333.1, 10.0.236184579.1

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 24 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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ $20$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
37Data not computed
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$
241Data not computed