# Properties

 Label 20.0.70900030741...9856.1 Degree $20$ Signature $[0, 10]$ Discriminant $2^{4}\cdot 83^{4}\cdot 983^{4}$ Root discriminant $11.03$ Ramified primes $2, 83, 983$ Class number $1$ Class group Trivial Galois Group 20T669

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 15, -37, 78, -139, 219, -307, 386, -442, 463, -442, 386, -307, 219, -139, 78, -37, 15, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 15*x^18 - 37*x^17 + 78*x^16 - 139*x^15 + 219*x^14 - 307*x^13 + 386*x^12 - 442*x^11 + 463*x^10 - 442*x^9 + 386*x^8 - 307*x^7 + 219*x^6 - 139*x^5 + 78*x^4 - 37*x^3 + 15*x^2 - 5*x + 1)
gp: K = bnfinit(x^20 - 5*x^19 + 15*x^18 - 37*x^17 + 78*x^16 - 139*x^15 + 219*x^14 - 307*x^13 + 386*x^12 - 442*x^11 + 463*x^10 - 442*x^9 + 386*x^8 - 307*x^7 + 219*x^6 - 139*x^5 + 78*x^4 - 37*x^3 + 15*x^2 - 5*x + 1, 1)

## Normalizeddefining polynomial

$$x^{20}$$ $$\mathstrut -\mathstrut 5 x^{19}$$ $$\mathstrut +\mathstrut 15 x^{18}$$ $$\mathstrut -\mathstrut 37 x^{17}$$ $$\mathstrut +\mathstrut 78 x^{16}$$ $$\mathstrut -\mathstrut 139 x^{15}$$ $$\mathstrut +\mathstrut 219 x^{14}$$ $$\mathstrut -\mathstrut 307 x^{13}$$ $$\mathstrut +\mathstrut 386 x^{12}$$ $$\mathstrut -\mathstrut 442 x^{11}$$ $$\mathstrut +\mathstrut 463 x^{10}$$ $$\mathstrut -\mathstrut 442 x^{9}$$ $$\mathstrut +\mathstrut 386 x^{8}$$ $$\mathstrut -\mathstrut 307 x^{7}$$ $$\mathstrut +\mathstrut 219 x^{6}$$ $$\mathstrut -\mathstrut 139 x^{5}$$ $$\mathstrut +\mathstrut 78 x^{4}$$ $$\mathstrut -\mathstrut 37 x^{3}$$ $$\mathstrut +\mathstrut 15 x^{2}$$ $$\mathstrut -\mathstrut 5 x$$ $$\mathstrut +\mathstrut 1$$

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: $$709000307415298179856=2^{4}\cdot 83^{4}\cdot 983^{4}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $11.03$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 83, 983$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{62} a^{18} - \frac{9}{31} a^{17} - \frac{19}{62} a^{15} + \frac{15}{62} a^{14} - \frac{5}{62} a^{13} + \frac{21}{62} a^{12} - \frac{17}{62} a^{11} + \frac{14}{31} a^{10} + \frac{17}{62} a^{9} + \frac{14}{31} a^{8} - \frac{17}{62} a^{7} + \frac{21}{62} a^{6} - \frac{5}{62} a^{5} + \frac{15}{62} a^{4} - \frac{19}{62} a^{3} - \frac{9}{31} a + \frac{1}{62}$, $\frac{1}{3782} a^{19} - \frac{3}{1891} a^{18} - \frac{325}{1891} a^{17} + \frac{1345}{3782} a^{16} - \frac{1267}{3782} a^{15} - \frac{1251}{3782} a^{14} - \frac{1031}{3782} a^{13} + \frac{297}{3782} a^{12} + \frac{563}{1891} a^{11} - \frac{1507}{3782} a^{10} - \frac{845}{1891} a^{9} + \frac{1187}{3782} a^{8} - \frac{679}{3782} a^{7} + \frac{433}{3782} a^{6} + \frac{823}{3782} a^{5} - \frac{1389}{3782} a^{4} - \frac{517}{1891} a^{3} - \frac{691}{1891} a^{2} + \frac{1397}{3782} a - \frac{335}{1891}$

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: $$\frac{2634}{1891} a^{19} - \frac{12571}{1891} a^{18} + \frac{37502}{1891} a^{17} - \frac{93663}{1891} a^{16} + \frac{197977}{1891} a^{15} - \frac{353409}{1891} a^{14} + \frac{562303}{1891} a^{13} - \frac{793088}{1891} a^{12} + \frac{1001013}{1891} a^{11} - \frac{1152092}{1891} a^{10} + \frac{1206534}{1891} a^{9} - \frac{1149237}{1891} a^{8} + \frac{1002508}{1891} a^{7} - \frac{792263}{1891} a^{6} + \frac{17923}{61} a^{5} - \frac{348156}{1891} a^{4} + \frac{187669}{1891} a^{3} - \frac{81326}{1891} a^{2} + \frac{30495}{1891} a - \frac{8590}{1891}$$,  $$\frac{599}{1891} a^{19} - \frac{2008}{1891} a^{18} + \frac{3795}{1891} a^{17} - \frac{7475}{1891} a^{16} + \frac{12717}{1891} a^{15} - \frac{14543}{1891} a^{14} + \frac{15550}{1891} a^{13} - \frac{15711}{1891} a^{12} + \frac{12136}{1891} a^{11} - \frac{13008}{1891} a^{10} + \frac{14991}{1891} a^{9} - \frac{14216}{1891} a^{8} + \frac{16375}{1891} a^{7} - \frac{21233}{1891} a^{6} + \frac{17970}{1891} a^{5} - \frac{17783}{1891} a^{4} + \frac{16132}{1891} a^{3} - \frac{9015}{1891} a^{2} + \frac{4580}{1891} a - \frac{2634}{1891}$$,  $$\frac{5240}{1891} a^{19} - \frac{24730}{1891} a^{18} + \frac{71793}{1891} a^{17} - \frac{173929}{1891} a^{16} + \frac{358718}{1891} a^{15} - \frac{622746}{1891} a^{14} + \frac{959372}{1891} a^{13} - \frac{1307609}{1891} a^{12} + \frac{1595714}{1891} a^{11} - \frac{1774842}{1891} a^{10} + \frac{1797013}{1891} a^{9} - \frac{1653573}{1891} a^{8} + \frac{1388286}{1891} a^{7} - \frac{1050700}{1891} a^{6} + \frac{703089}{1891} a^{5} - \frac{413603}{1891} a^{4} + \frac{208663}{1891} a^{3} - \frac{84245}{1891} a^{2} + \frac{30719}{1891} a - \frac{7631}{1891}$$,  $$\frac{1183}{3782} a^{19} - \frac{2523}{3782} a^{18} + \frac{1716}{1891} a^{17} - \frac{4869}{3782} a^{16} - \frac{564}{1891} a^{15} + \frac{12926}{1891} a^{14} - \frac{31282}{1891} a^{13} + \frac{64869}{1891} a^{12} - \frac{213133}{3782} a^{11} + \frac{289267}{3782} a^{10} - \frac{355747}{3782} a^{9} + \frac{390157}{3782} a^{8} - \frac{187122}{1891} a^{7} + \frac{168005}{1891} a^{6} - \frac{133535}{1891} a^{5} + \frac{90143}{1891} a^{4} - \frac{107471}{3782} a^{3} + \frac{27824}{1891} a^{2} - \frac{18139}{3782} a + \frac{6185}{3782}$$,  $$\frac{3301}{3782} a^{19} - \frac{18037}{3782} a^{18} + \frac{26944}{1891} a^{17} - \frac{132593}{3782} a^{16} + \frac{140410}{1891} a^{15} - \frac{249385}{1891} a^{14} + \frac{389140}{1891} a^{13} - \frac{544514}{1891} a^{12} + \frac{1349577}{3782} a^{11} - \frac{1528839}{3782} a^{10} + \frac{1584235}{3782} a^{9} - \frac{1485825}{3782} a^{8} + \frac{638032}{1891} a^{7} - \frac{497800}{1891} a^{6} + \frac{342254}{1891} a^{5} - \frac{208632}{1891} a^{4} + \frac{225477}{3782} a^{3} - \frac{47720}{1891} a^{2} + \frac{33691}{3782} a - \frac{12559}{3782}$$,  $$\frac{936}{1891} a^{19} - \frac{7145}{3782} a^{18} + \frac{10994}{1891} a^{17} - \frac{26960}{1891} a^{16} + \frac{107155}{3782} a^{15} - \frac{185339}{3782} a^{14} + \frac{288477}{3782} a^{13} - \frac{375547}{3782} a^{12} + \frac{453735}{3782} a^{11} - \frac{245208}{1891} a^{10} + \frac{472225}{3782} a^{9} - \frac{210289}{1891} a^{8} + \frac{334859}{3782} a^{7} - \frac{226851}{3782} a^{6} + \frac{136009}{3782} a^{5} - \frac{69257}{3782} a^{4} + \frac{17633}{3782} a^{3} - \frac{108}{1891} a^{2} + \frac{57}{1891} a + \frac{5471}{3782}$$,  $$\frac{197}{62} a^{19} - \frac{461}{31} a^{18} + \frac{1308}{31} a^{17} - \frac{6285}{62} a^{16} + \frac{12929}{62} a^{15} - \frac{22261}{62} a^{14} + \frac{34107}{62} a^{13} - \frac{46515}{62} a^{12} + \frac{28349}{31} a^{11} - \frac{63237}{62} a^{10} + \frac{32223}{31} a^{9} - \frac{59581}{62} a^{8} + \frac{50483}{62} a^{7} - \frac{38757}{62} a^{6} + \frac{26415}{62} a^{5} - \frac{15905}{62} a^{4} + \frac{4243}{31} a^{3} - \frac{1835}{31} a^{2} + \frac{1387}{62} a - \frac{207}{31}$$,  $$\frac{4413}{1891} a^{19} - \frac{45575}{3782} a^{18} + \frac{66138}{1891} a^{17} - \frac{159208}{1891} a^{16} + \frac{21367}{122} a^{15} - \frac{1154141}{3782} a^{14} + \frac{1766995}{3782} a^{13} - \frac{2423927}{3782} a^{12} + \frac{2955833}{3782} a^{11} - \frac{1645585}{1891} a^{10} + \frac{3351791}{3782} a^{9} - \frac{1545457}{1891} a^{8} + \frac{2595397}{3782} a^{7} - \frac{1983773}{3782} a^{6} + \frac{1334537}{3782} a^{5} - \frac{787471}{3782} a^{4} + \frac{408033}{3782} a^{3} - \frac{85386}{1891} a^{2} + \frac{28422}{1891} a - \frac{21247}{3782}$$,  $$\frac{7059}{3782} a^{19} - \frac{13979}{1891} a^{18} + \frac{36452}{1891} a^{17} - \frac{168655}{3782} a^{16} + \frac{332273}{3782} a^{15} - \frac{17429}{122} a^{14} + \frac{800401}{3782} a^{13} - \frac{1046563}{3782} a^{12} + \frac{618241}{1891} a^{11} - \frac{1343343}{3782} a^{10} + \frac{668224}{1891} a^{9} - \frac{1198577}{3782} a^{8} + \frac{994497}{3782} a^{7} - \frac{744607}{3782} a^{6} + \frac{491943}{3782} a^{5} - \frac{292855}{3782} a^{4} + \frac{77048}{1891} a^{3} - \frac{31136}{1891} a^{2} + \frac{26271}{3782} a - \frac{1381}{1891}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$103.364702597$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A non-solvable group of order 61440 The 126 conjugacy class representatives for t20n669 are not computed Character table for t20n669 is not computed

## Intermediate fields

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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$V_4$$[2]^{2} 2.8.0.1x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
83Data not computed
983Data not computed