# Properties

 Label 20.0.40542497504...9161.1 Degree $20$ Signature $[0, 10]$ Discriminant $7^{2}\cdot 53^{6}\cdot 139^{4}$ Root discriminant $10.72$ Ramified primes $7, 53, 139$ Class number $1$ Class group Trivial Galois Group 20T799

# Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 25, -76, 180, -349, 580, -845, 1093, -1265, 1317, -1233, 1036, -778, 519, -304, 154, -66, 23, -6, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 23*x^18 - 66*x^17 + 154*x^16 - 304*x^15 + 519*x^14 - 778*x^13 + 1036*x^12 - 1233*x^11 + 1317*x^10 - 1265*x^9 + 1093*x^8 - 845*x^7 + 580*x^6 - 349*x^5 + 180*x^4 - 76*x^3 + 25*x^2 - 6*x + 1)
gp: K = bnfinit(x^20 - 6*x^19 + 23*x^18 - 66*x^17 + 154*x^16 - 304*x^15 + 519*x^14 - 778*x^13 + 1036*x^12 - 1233*x^11 + 1317*x^10 - 1265*x^9 + 1093*x^8 - 845*x^7 + 580*x^6 - 349*x^5 + 180*x^4 - 76*x^3 + 25*x^2 - 6*x + 1, 1)

## Normalizeddefining polynomial

$$x^{20}$$ $$\mathstrut -\mathstrut 6 x^{19}$$ $$\mathstrut +\mathstrut 23 x^{18}$$ $$\mathstrut -\mathstrut 66 x^{17}$$ $$\mathstrut +\mathstrut 154 x^{16}$$ $$\mathstrut -\mathstrut 304 x^{15}$$ $$\mathstrut +\mathstrut 519 x^{14}$$ $$\mathstrut -\mathstrut 778 x^{13}$$ $$\mathstrut +\mathstrut 1036 x^{12}$$ $$\mathstrut -\mathstrut 1233 x^{11}$$ $$\mathstrut +\mathstrut 1317 x^{10}$$ $$\mathstrut -\mathstrut 1265 x^{9}$$ $$\mathstrut +\mathstrut 1093 x^{8}$$ $$\mathstrut -\mathstrut 845 x^{7}$$ $$\mathstrut +\mathstrut 580 x^{6}$$ $$\mathstrut -\mathstrut 349 x^{5}$$ $$\mathstrut +\mathstrut 180 x^{4}$$ $$\mathstrut -\mathstrut 76 x^{3}$$ $$\mathstrut +\mathstrut 25 x^{2}$$ $$\mathstrut -\mathstrut 6 x$$ $$\mathstrut +\mathstrut 1$$

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: $$405424975045226129161=7^{2}\cdot 53^{6}\cdot 139^{4}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $10.72$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $7, 53, 139$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{193} a^{19} - \frac{70}{193} a^{18} + \frac{64}{193} a^{17} + \frac{84}{193} a^{16} - \frac{11}{193} a^{15} + \frac{14}{193} a^{14} + \frac{9}{193} a^{13} - \frac{3}{193} a^{12} + \frac{70}{193} a^{11} + \frac{77}{193} a^{10} + \frac{56}{193} a^{9} - \frac{24}{193} a^{8} - \frac{73}{193} a^{7} - \frac{33}{193} a^{6} - \frac{10}{193} a^{5} - \frac{95}{193} a^{4} + \frac{84}{193} a^{3} - \frac{48}{193} a^{2} + \frac{9}{193} a - \frac{3}{193}$

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: $$\frac{784}{193} a^{19} - \frac{4507}{193} a^{18} + \frac{16594}{193} a^{17} - \frac{46084}{193} a^{16} + \frac{103895}{193} a^{15} - \frac{198429}{193} a^{14} + \frac{327243}{193} a^{13} - \frac{473272}{193} a^{12} + \frac{607439}{193} a^{11} - \frac{695420}{193} a^{10} + \frac{712649}{193} a^{9} - \frac{655137}{193} a^{8} + \frac{539524}{193} a^{7} - \frac{394888}{193} a^{6} + \frac{253868}{193} a^{5} - \frac{141258}{193} a^{4} + \frac{65277}{193} a^{3} - \frac{23350}{193} a^{2} + \frac{5898}{193} a - \frac{1001}{193}$$,  $$\frac{266}{193} a^{19} - \frac{1636}{193} a^{18} + \frac{6216}{193} a^{17} - \frac{17800}{193} a^{16} + \frac{41271}{193} a^{15} - \frac{81003}{193} a^{14} + \frac{137301}{193} a^{13} - \frac{204220}{193} a^{12} + \frac{269520}{193} a^{11} - \frac{317654}{193} a^{10} + \frac{335276}{193} a^{9} - \frac{317693}{193} a^{8} + \frac{270082}{193} a^{7} - \frac{204673}{193} a^{6} + \frac{136686}{193} a^{5} - \frac{79696}{193} a^{4} + \frac{39135}{193} a^{3} - \frac{15277}{193} a^{2} + \frac{4324}{193} a - \frac{991}{193}$$,  $$\frac{1001}{193} a^{19} - \frac{5222}{193} a^{18} + \frac{18516}{193} a^{17} - \frac{49472}{193} a^{16} + \frac{108070}{193} a^{15} - \frac{200409}{193} a^{14} + \frac{321090}{193} a^{13} - \frac{451535}{193} a^{12} + \frac{563764}{193} a^{11} - \frac{626794}{193} a^{10} + \frac{622897}{193} a^{9} - \frac{553616}{193} a^{8} + \frac{438956}{193} a^{7} - \frac{306321}{193} a^{6} + \frac{185692}{193} a^{5} - \frac{95481}{193} a^{4} + \frac{38922}{193} a^{3} - \frac{10799}{193} a^{2} + \frac{1675}{193} a - \frac{108}{193}$$,  $$\frac{156}{193} a^{19} - \frac{1270}{193} a^{18} + \frac{5159}{193} a^{17} - \frac{15653}{193} a^{16} + \frac{37463}{193} a^{15} - \frac{75209}{193} a^{14} + \frac{129363}{193} a^{13} - \frac{193854}{193} a^{12} + \frac{256416}{193} a^{11} - \frac{301806}{193} a^{10} + \frac{317343}{193} a^{9} - \frac{299034}{193} a^{8} + \frac{252250}{193} a^{7} - \frac{190042}{193} a^{6} + \frac{125820}{193} a^{5} - \frac{72527}{193} a^{4} + \frac{35106}{193} a^{3} - \frac{13664}{193} a^{2} + \frac{3913}{193} a - \frac{854}{193}$$,  $$\frac{230}{193} a^{19} - \frac{1046}{193} a^{18} + \frac{3719}{193} a^{17} - \frac{9630}{193} a^{16} + \frac{20823}{193} a^{15} - \frac{38082}{193} a^{14} + \frac{60163}{193} a^{13} - \frac{83487}{193} a^{12} + \frac{102371}{193} a^{11} - \frac{111214}{193} a^{10} + \frac{107643}{193} a^{9} - \frac{91984}{193} a^{8} + \frac{69481}{193} a^{7} - \frac{45032}{193} a^{6} + \frac{24527}{193} a^{5} - \frac{10077}{193} a^{4} + \frac{2336}{193} a^{3} + \frac{540}{193} a^{2} - \frac{825}{193} a + \frac{275}{193}$$,  $$\frac{26}{193} a^{19} - \frac{276}{193} a^{18} + \frac{1085}{193} a^{17} - \frac{3606}{193} a^{16} + \frac{8978}{193} a^{15} - \frac{19129}{193} a^{14} + \frac{34588}{193} a^{13} - \frac{54504}{193} a^{12} + \frac{75739}{193} a^{11} - \frac{93147}{193} a^{10} + \frac{102009}{193} a^{9} - \frac{100212}{193} a^{8} + \frac{87654}{193} a^{7} - \frac{68408}{193} a^{6} + \frac{46832}{193} a^{5} - \frac{28332}{193} a^{4} + \frac{13957}{193} a^{3} - \frac{5687}{193} a^{2} + \frac{1778}{193} a - \frac{464}{193}$$,  $$\frac{564}{193} a^{19} - \frac{2617}{193} a^{18} + \frac{8883}{193} a^{17} - \frac{22490}{193} a^{16} + \frac{47064}{193} a^{15} - \frac{83586}{193} a^{14} + \frac{128403}{193} a^{13} - \frac{173269}{193} a^{12} + \frac{207969}{193} a^{11} - \frac{221947}{193} a^{10} + \frac{211846}{193} a^{9} - \frac{180481}{193} a^{8} + \frac{136967}{193} a^{7} - \frac{90215}{193} a^{6} + \frac{51488}{193} a^{5} - \frac{24051}{193} a^{4} + \frac{8390}{193} a^{3} - \frac{1403}{193} a^{2} + \frac{58}{193} a + \frac{238}{193}$$,  $$\frac{970}{193} a^{19} - \frac{5561}{193} a^{18} + \frac{20392}{193} a^{17} - \frac{56515}{193} a^{16} + \frac{127132}{193} a^{15} - \frac{242338}{193} a^{14} + \frac{398976}{193} a^{13} - \frac{575927}{193} a^{12} + \frac{737996}{193} a^{11} - \frac{843218}{193} a^{10} + \frac{862218}{193} a^{9} - \frac{790648}{193} a^{8} + \frac{649080}{193} a^{7} - \frac{473015}{193} a^{6} + \frac{302381}{193} a^{5} - \frac{167034}{193} a^{4} + \frac{76269}{193} a^{3} - \frac{26681}{193} a^{2} + \frac{6800}{193} a - \frac{1173}{193}$$,  $$\frac{690}{193} a^{19} - \frac{3717}{193} a^{18} + \frac{13280}{193} a^{17} - \frac{35838}{193} a^{16} + \frac{78681}{193} a^{15} - \frac{146670}{193} a^{14} + \frac{235880}{193} a^{13} - \frac{332872}{193} a^{12} + \frac{416930}{193} a^{11} - \frac{465268}{193} a^{10} + \frac{464591}{193} a^{9} - \frac{415491}{193} a^{8} + \frac{332156}{193} a^{7} - \frac{235070}{193} a^{6} + \frac{145184}{193} a^{5} - \frac{77130}{193} a^{4} + \frac{33256}{193} a^{3} - \frac{10732}{193} a^{2} + \frac{2350}{193} a - \frac{333}{193}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$62.9466310057$$ 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 122880 The 252 conjugacy class representatives for t20n799 are not computed Character table for t20n799 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2} 7.2.0.1x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$V_4$$[\ ]_{2}^{2} 7.6.0.1x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6} 5353.4.0.1x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$V_4$$[\ ]_{2}^{2} 53.4.2.1x^{4} + 477 x^{2} + 70225$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$V_4$$[\ ]_{2}^{2} 53.4.0.1x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
$139$$\Q_{139}$$x + 4$$1$$1$$0Trivial[\ ] \Q_{139}$$x + 4$$1$$1$$0Trivial[\ ] \Q_{139}$$x + 4$$1$$1$$0Trivial[\ ] \Q_{139}$$x + 4$$1$$1$$0Trivial[\ ] 139.4.0.1x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$V_4$$[\ ]_{2}^{2} 139.4.2.1x^{4} + 417 x^{2} + 77284$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
139.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$