Properties

Label 20.0.75066106602...9776.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{8}\cdot 41381^{4}$
Root discriminant $11.06$
Ramified primes $2, 41381$
Class number $1$
Class group Trivial
Galois Group 20T673

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Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

\(x^{20} \) \(\mathstrut +\mathstrut x^{16} \) \(\mathstrut +\mathstrut 5 x^{12} \) \(\mathstrut +\mathstrut 10 x^{10} \) \(\mathstrut +\mathstrut 9 x^{8} \) \(\mathstrut +\mathstrut 7 x^{6} \) \(\mathstrut +\mathstrut 3 x^{4} \) \(\mathstrut +\mathstrut 4 x^{2} \) \(\mathstrut +\mathstrut 1 \)

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:  \(750661066024355819776=2^{8}\cdot 41381^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.06$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 41381$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{8326} a^{18} + \frac{1123}{8326} a^{16} - \frac{259}{8326} a^{14} - \frac{1805}{4163} a^{12} - \frac{1}{2} a^{11} + \frac{737}{8326} a^{10} + \frac{3387}{8326} a^{8} - \frac{686}{4163} a^{6} - \frac{1}{2} a^{5} + \frac{1862}{4163} a^{4} - \frac{1}{2} a^{3} - \frac{880}{4163} a^{2} + \frac{949}{8326}$, $\frac{1}{8326} a^{19} + \frac{1123}{8326} a^{17} - \frac{259}{8326} a^{15} - \frac{1805}{4163} a^{13} - \frac{1}{2} a^{12} + \frac{737}{8326} a^{11} + \frac{3387}{8326} a^{9} - \frac{686}{4163} a^{7} - \frac{1}{2} a^{6} + \frac{1862}{4163} a^{5} - \frac{1}{2} a^{4} - \frac{880}{4163} a^{3} + \frac{949}{8326} a$

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:  \( \frac{2032}{4163} a^{19} + \frac{612}{4163} a^{17} + \frac{2413}{4163} a^{15} - \frac{314}{4163} a^{13} + \frac{11393}{4163} a^{11} + \frac{21760}{4163} a^{9} + \frac{26284}{4163} a^{7} + \frac{19649}{4163} a^{5} + \frac{8023}{4163} a^{3} + \frac{9225}{4163} a \),  \( a \),  \( \frac{4545}{8326} a^{19} - \frac{611}{8326} a^{18} + \frac{197}{8326} a^{17} + \frac{371}{4163} a^{16} + \frac{5137}{8326} a^{15} + \frac{55}{8326} a^{14} - \frac{1067}{8326} a^{13} - \frac{340}{4163} a^{12} + \frac{11714}{4163} a^{11} - \frac{703}{8326} a^{10} + \frac{22467}{4163} a^{9} - \frac{4609}{8326} a^{8} + \frac{46227}{8326} a^{7} + \frac{2846}{4163} a^{6} + \frac{16063}{4163} a^{5} + \frac{1797}{8326} a^{4} + \frac{5206}{4163} a^{3} + \frac{653}{4163} a^{2} + \frac{6413}{4163} a - \frac{591}{4163} \),  \( \frac{4545}{8326} a^{19} + \frac{611}{8326} a^{18} + \frac{197}{8326} a^{17} - \frac{371}{4163} a^{16} + \frac{5137}{8326} a^{15} - \frac{55}{8326} a^{14} - \frac{1067}{8326} a^{13} + \frac{340}{4163} a^{12} + \frac{11714}{4163} a^{11} + \frac{703}{8326} a^{10} + \frac{22467}{4163} a^{9} + \frac{4609}{8326} a^{8} + \frac{46227}{8326} a^{7} - \frac{2846}{4163} a^{6} + \frac{16063}{4163} a^{5} - \frac{1797}{8326} a^{4} + \frac{5206}{4163} a^{3} - \frac{653}{4163} a^{2} + \frac{6413}{4163} a + \frac{591}{4163} \),  \( \frac{1632}{4163} a^{19} + \frac{1827}{8326} a^{18} - \frac{2131}{8326} a^{17} - \frac{319}{4163} a^{16} + \frac{1938}{4163} a^{15} + \frac{1389}{8326} a^{14} - \frac{875}{4163} a^{13} - \frac{639}{4163} a^{12} + \frac{8003}{4163} a^{11} + \frac{5088}{4163} a^{10} + \frac{11609}{4163} a^{9} + \frac{7160}{4163} a^{8} + \frac{4753}{4163} a^{7} + \frac{3904}{4163} a^{6} + \frac{11665}{8326} a^{5} + \frac{703}{4163} a^{4} + \frac{150}{4163} a^{3} - \frac{842}{4163} a^{2} + \frac{12753}{8326} a + \frac{10341}{8326} \),  \( \frac{1632}{4163} a^{19} - \frac{1827}{8326} a^{18} - \frac{2131}{8326} a^{17} + \frac{319}{4163} a^{16} + \frac{1938}{4163} a^{15} - \frac{1389}{8326} a^{14} - \frac{875}{4163} a^{13} + \frac{639}{4163} a^{12} + \frac{8003}{4163} a^{11} - \frac{5088}{4163} a^{10} + \frac{11609}{4163} a^{9} - \frac{7160}{4163} a^{8} + \frac{4753}{4163} a^{7} - \frac{3904}{4163} a^{6} + \frac{11665}{8326} a^{5} - \frac{703}{4163} a^{4} + \frac{150}{4163} a^{3} + \frac{842}{4163} a^{2} + \frac{12753}{8326} a - \frac{10341}{8326} \),  \( \frac{1590}{4163} a^{18} - \frac{357}{4163} a^{16} + \frac{327}{4163} a^{14} + \frac{877}{4163} a^{12} + \frac{6190}{4163} a^{10} + \frac{15060}{4163} a^{8} + \frac{4095}{4163} a^{6} + \frac{1374}{4163} a^{4} - \frac{864}{4163} a^{2} + \frac{1904}{4163} \),  \( \frac{104}{181} a^{19} - \frac{386}{4163} a^{18} - \frac{87}{362} a^{17} - \frac{526}{4163} a^{16} + \frac{247}{362} a^{15} + \frac{62}{4163} a^{14} - \frac{46}{181} a^{13} - \frac{1145}{4163} a^{12} + \frac{1075}{362} a^{11} - \frac{1398}{4163} a^{10} + \frac{1673}{362} a^{9} - \frac{12889}{8326} a^{8} + \frac{1147}{362} a^{7} - \frac{7435}{4163} a^{6} + \frac{499}{181} a^{5} - \frac{5392}{4163} a^{4} + \frac{445}{362} a^{3} - \frac{10907}{8326} a^{2} + \frac{645}{362} a - \frac{4103}{8326} \),  \( \frac{2018}{4163} a^{19} + \frac{1066}{4163} a^{18} - \frac{1079}{8326} a^{17} + \frac{511}{8326} a^{16} + \frac{1876}{4163} a^{15} + \frac{1491}{8326} a^{14} + \frac{270}{4163} a^{13} + \frac{867}{8326} a^{12} + \frac{9401}{4163} a^{11} + \frac{10159}{8326} a^{10} + \frac{36107}{8326} a^{9} + \frac{23257}{8326} a^{8} + \frac{12188}{4163} a^{7} + \frac{11150}{4163} a^{6} + \frac{22449}{8326} a^{5} + \frac{17379}{8326} a^{4} + \frac{11207}{8326} a^{3} + \frac{6869}{8326} a^{2} + \frac{8428}{4163} a + \frac{4213}{8326} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 91.2702526158 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

20T673:

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 t20n673 are not computed
Character table for t20n673 is not computed

Intermediate fields

5.1.41381.1, 10.2.1712387161.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: 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 R $16{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }$ $16{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }$ $16{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.12.0.1$x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$$1$$12$$0$$C_{12}$$[\ ]^{12}$
41381Data not computed