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

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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)

Normalized defining 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

20T799:

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

5.3.7367.1, 10.2.2876452517.1, 10.0.379908823.1, 10.4.20135167619.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ 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$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
$53$53.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
53.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
$139$$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
139.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
139.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$