Properties

Label 20.0.16073775233...1377.1
Degree $20$
Signature $[0, 10]$
Discriminant $28753\cdot 236438047^{2}$
Root discriminant $11.49$
Ramified primes $28753, 236438047$
Class number $1$
Class group Trivial
Galois Group 20T1110

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 3, -1, 2, 5, 0, 6, 3, 4, 5, 4, 4, 7, -1, 9, -3, 5, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 5*x^18 - 3*x^17 + 9*x^16 - x^15 + 7*x^14 + 4*x^13 + 4*x^12 + 5*x^11 + 4*x^10 + 3*x^9 + 6*x^8 + 5*x^6 + 2*x^5 - x^4 + 3*x^3 + 1)
gp: K = bnfinit(x^20 - x^19 + 5*x^18 - 3*x^17 + 9*x^16 - x^15 + 7*x^14 + 4*x^13 + 4*x^12 + 5*x^11 + 4*x^10 + 3*x^9 + 6*x^8 + 5*x^6 + 2*x^5 - x^4 + 3*x^3 + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut x^{19} \) \(\mathstrut +\mathstrut 5 x^{18} \) \(\mathstrut -\mathstrut 3 x^{17} \) \(\mathstrut +\mathstrut 9 x^{16} \) \(\mathstrut -\mathstrut x^{15} \) \(\mathstrut +\mathstrut 7 x^{14} \) \(\mathstrut +\mathstrut 4 x^{13} \) \(\mathstrut +\mathstrut 4 x^{12} \) \(\mathstrut +\mathstrut 5 x^{11} \) \(\mathstrut +\mathstrut 4 x^{10} \) \(\mathstrut +\mathstrut 3 x^{9} \) \(\mathstrut +\mathstrut 6 x^{8} \) \(\mathstrut +\mathstrut 5 x^{6} \) \(\mathstrut +\mathstrut 2 x^{5} \) \(\mathstrut -\mathstrut x^{4} \) \(\mathstrut +\mathstrut 3 x^{3} \) \(\mathstrut +\mathstrut 1 \)

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:  \(1607377523338966031377=28753\cdot 236438047^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.49$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $28753, 236438047$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{503} a^{19} + \frac{201}{503} a^{18} - \frac{136}{503} a^{17} + \frac{190}{503} a^{16} + \frac{161}{503} a^{15} - \frac{174}{503} a^{14} + \frac{69}{503} a^{13} - \frac{142}{503} a^{12} - \frac{9}{503} a^{11} + \frac{199}{503} a^{10} - \frac{38}{503} a^{9} - \frac{128}{503} a^{8} - \frac{197}{503} a^{7} - \frac{57}{503} a^{6} + \frac{60}{503} a^{5} + \frac{50}{503} a^{4} + \frac{39}{503} a^{3} - \frac{167}{503} a^{2} - \frac{33}{503} a - \frac{127}{503}$

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:  \( \frac{30}{503} a^{19} - \frac{509}{503} a^{18} + \frac{447}{503} a^{17} - \frac{2348}{503} a^{16} + \frac{806}{503} a^{15} - \frac{4214}{503} a^{14} - \frac{948}{503} a^{13} - \frac{4260}{503} a^{12} - \frac{3288}{503} a^{11} - \frac{4090}{503} a^{10} - \frac{3655}{503} a^{9} - \frac{3840}{503} a^{8} - \frac{2389}{503} a^{7} - \frac{3722}{503} a^{6} - \frac{715}{503} a^{5} - \frac{2524}{503} a^{4} - \frac{1345}{503} a^{3} + \frac{20}{503} a^{2} - \frac{990}{503} a - \frac{289}{503} \),  \( \frac{289}{503} a^{19} - \frac{259}{503} a^{18} + \frac{936}{503} a^{17} - \frac{420}{503} a^{16} + \frac{253}{503} a^{15} + \frac{517}{503} a^{14} - \frac{2191}{503} a^{13} + \frac{208}{503} a^{12} - \frac{3104}{503} a^{11} - \frac{1843}{503} a^{10} - \frac{2934}{503} a^{9} - \frac{2788}{503} a^{8} - \frac{2106}{503} a^{7} - \frac{2389}{503} a^{6} - \frac{2277}{503} a^{5} - \frac{137}{503} a^{4} - \frac{2813}{503} a^{3} - \frac{478}{503} a^{2} + \frac{20}{503} a - \frac{990}{503} \),  \( \frac{776}{503} a^{19} - \frac{960}{503} a^{18} + \frac{3615}{503} a^{17} - \frac{2957}{503} a^{16} + \frac{5725}{503} a^{15} - \frac{2232}{503} a^{14} + \frac{3244}{503} a^{13} - \frac{35}{503} a^{12} + \frac{1064}{503} a^{11} + \frac{3}{503} a^{10} + \frac{692}{503} a^{9} - \frac{740}{503} a^{8} + \frac{2555}{503} a^{7} - \frac{2483}{503} a^{6} + \frac{1793}{503} a^{5} + \frac{69}{503} a^{4} - \frac{1928}{503} a^{3} + \frac{685}{503} a^{2} + \frac{45}{503} a - \frac{467}{503} \),  \( \frac{442}{503} a^{19} - \frac{692}{503} a^{18} + \frac{2260}{503} a^{17} - \frac{2033}{503} a^{16} + \frac{3760}{503} a^{15} - \frac{955}{503} a^{14} + \frac{1827}{503} a^{13} + \frac{1620}{503} a^{12} + \frac{46}{503} a^{11} + \frac{1442}{503} a^{10} - \frac{197}{503} a^{9} - \frac{240}{503} a^{8} + \frac{951}{503} a^{7} - \frac{2056}{503} a^{6} + \frac{867}{503} a^{5} - \frac{32}{503} a^{4} - \frac{1876}{503} a^{3} + \frac{1133}{503} a^{2} - \frac{502}{503} a - \frac{804}{503} \),  \( \frac{230}{503} a^{19} - \frac{46}{503} a^{18} + \frac{912}{503} a^{17} + \frac{442}{503} a^{16} + \frac{1317}{503} a^{15} + \frac{2232}{503} a^{14} + \frac{1283}{503} a^{13} + \frac{3556}{503} a^{12} + \frac{1954}{503} a^{11} + \frac{3518}{503} a^{10} + \frac{2326}{503} a^{9} + \frac{2752}{503} a^{8} + \frac{2475}{503} a^{7} + \frac{1980}{503} a^{6} + \frac{1225}{503} a^{5} + \frac{2446}{503} a^{4} + \frac{419}{503} a^{3} + \frac{824}{503} a^{2} + \frac{961}{503} a - \frac{36}{503} \),  \( \frac{619}{503} a^{19} - \frac{828}{503} a^{18} + \frac{2835}{503} a^{17} - \frac{2607}{503} a^{16} + \frac{4592}{503} a^{15} - \frac{2076}{503} a^{14} + \frac{2974}{503} a^{13} + \frac{127}{503} a^{12} + \frac{1471}{503} a^{11} + \frac{952}{503} a^{10} + \frac{1125}{503} a^{9} + \frac{242}{503} a^{8} + \frac{2298}{503} a^{7} - \frac{1582}{503} a^{6} + \frac{1427}{503} a^{5} + \frac{267}{503} a^{4} - \frac{1512}{503} a^{3} + \frac{748}{503} a^{2} + \frac{196}{503} a - \frac{145}{503} \),  \( \frac{414}{503} a^{19} - \frac{284}{503} a^{18} + \frac{2044}{503} a^{17} - \frac{311}{503} a^{16} + \frac{3779}{503} a^{15} + \frac{1905}{503} a^{14} + \frac{3919}{503} a^{13} + \frac{4590}{503} a^{12} + \frac{4322}{503} a^{11} + \frac{4924}{503} a^{10} + \frac{4388}{503} a^{9} + \frac{3847}{503} a^{8} + \frac{4455}{503} a^{7} + \frac{2055}{503} a^{6} + \frac{3211}{503} a^{5} + \frac{2592}{503} a^{4} + \frac{553}{503} a^{3} + \frac{1282}{503} a^{2} + \frac{925}{503} a - \frac{266}{503} \),  \( \frac{413}{503} a^{19} - \frac{485}{503} a^{18} + \frac{1677}{503} a^{17} - \frac{1507}{503} a^{16} + \frac{2109}{503} a^{15} - \frac{1442}{503} a^{14} + \frac{329}{503} a^{13} - \frac{1304}{503} a^{12} - \frac{699}{503} a^{11} - \frac{1814}{503} a^{10} - \frac{604}{503} a^{9} - \frac{2061}{503} a^{8} + \frac{628}{503} a^{7} - \frac{2415}{503} a^{6} + \frac{133}{503} a^{5} - \frac{476}{503} a^{4} - \frac{1498}{503} a^{3} - \frac{60}{503} a^{2} + \frac{455}{503} a - \frac{139}{503} \),  \( \frac{198}{503} a^{19} - \frac{442}{503} a^{18} + \frac{737}{503} a^{17} - \frac{1111}{503} a^{16} + \frac{189}{503} a^{15} - \frac{248}{503} a^{14} - \frac{1931}{503} a^{13} + \frac{555}{503} a^{12} - \frac{1782}{503} a^{11} - \frac{335}{503} a^{10} - \frac{482}{503} a^{9} - \frac{697}{503} a^{8} + \frac{731}{503} a^{7} - \frac{1226}{503} a^{6} + \frac{311}{503} a^{5} + \frac{846}{503} a^{4} - \frac{1332}{503} a^{3} + \frac{635}{503} a^{2} + \frac{508}{503} a - \frac{499}{503} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 140.247818215 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

20T1110:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 3715891200
The 481 conjugacy class representatives for t20n1110 are not computed
Character table for t20n1110 is not computed

Intermediate fields

10.0.236438047.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ $20$ $16{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ $16{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $18{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
28753Data not computed
236438047Data not computed