Properties

Label 20.2.22659664318...5904.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{10}\cdot 38569^{4}$
Root discriminant $11.69$
Ramified primes $2, 38569$
Class number $1$
Class group Trivial
Galois Group 20T992

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 3, 0, -4, 0, -13, 0, 3, 0, 21, 0, -1, 0, -7, 0, -4, 0, 1, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + x^18 - 4*x^16 - 7*x^14 - x^12 + 21*x^10 + 3*x^8 - 13*x^6 - 4*x^4 + 3*x^2 - 1)
gp: K = bnfinit(x^20 + x^18 - 4*x^16 - 7*x^14 - x^12 + 21*x^10 + 3*x^8 - 13*x^6 - 4*x^4 + 3*x^2 - 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut +\mathstrut x^{18} \) \(\mathstrut -\mathstrut 4 x^{16} \) \(\mathstrut -\mathstrut 7 x^{14} \) \(\mathstrut -\mathstrut x^{12} \) \(\mathstrut +\mathstrut 21 x^{10} \) \(\mathstrut +\mathstrut 3 x^{8} \) \(\mathstrut -\mathstrut 13 x^{6} \) \(\mathstrut -\mathstrut 4 x^{4} \) \(\mathstrut +\mathstrut 3 x^{2} \) \(\mathstrut -\mathstrut 1 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $20$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[2, 9]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-2265966431812150395904=-\,2^{10}\cdot 38569^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.69$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 38569$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{48194} a^{18} - \frac{4192}{24097} a^{16} - \frac{7605}{24097} a^{14} - \frac{1}{2} a^{13} - \frac{4789}{24097} a^{12} - \frac{1886}{24097} a^{10} - \frac{1}{2} a^{9} + \frac{12977}{48194} a^{8} - \frac{1}{2} a^{7} + \frac{4955}{24097} a^{6} - \frac{1}{2} a^{5} + \frac{7595}{24097} a^{4} - \frac{1}{2} a^{3} - \frac{15509}{48194} a^{2} + \frac{7778}{24097}$, $\frac{1}{48194} a^{19} - \frac{4192}{24097} a^{17} + \frac{8887}{48194} a^{15} + \frac{14519}{48194} a^{13} - \frac{1}{2} a^{12} + \frac{20325}{48194} a^{11} - \frac{1}{2} a^{10} + \frac{12977}{48194} a^{9} - \frac{1}{2} a^{8} + \frac{4955}{24097} a^{7} + \frac{7595}{24097} a^{5} + \frac{4294}{24097} a^{3} + \frac{7778}{24097} a - \frac{1}{2}$

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:  $10$
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:  \( \frac{7500}{24097} a^{18} + \frac{13170}{24097} a^{16} - \frac{23899}{24097} a^{14} - \frac{74134}{24097} a^{12} - \frac{48316}{24097} a^{10} + \frac{144299}{24097} a^{8} + \frac{130337}{24097} a^{6} - \frac{77907}{24097} a^{4} - \frac{73572}{24097} a^{2} + \frac{16423}{24097} \),  \( a \),  \( \frac{3639}{24097} a^{18} - \frac{2574}{24097} a^{16} - \frac{22478}{24097} a^{14} - \frac{10080}{24097} a^{12} + \frac{33079}{24097} a^{10} + \frac{113668}{24097} a^{8} - \frac{58913}{24097} a^{6} - \frac{50302}{24097} a^{4} - \frac{26174}{24097} a^{2} + \frac{4431}{24097} \),  \( \frac{18581}{48194} a^{19} - \frac{14833}{48194} a^{18} + \frac{14049}{24097} a^{17} - \frac{14421}{24097} a^{16} - \frac{27794}{24097} a^{15} + \frac{37913}{48194} a^{14} - \frac{157055}{48194} a^{13} + \frac{69572}{24097} a^{12} - \frac{54922}{24097} a^{11} + \frac{117327}{48194} a^{10} + \frac{162158}{24097} a^{9} - \frac{108939}{24097} a^{8} + \frac{205309}{48194} a^{7} - \frac{220203}{48194} a^{6} - \frac{99159}{48194} a^{5} + \frac{114165}{48194} a^{4} - \frac{94741}{24097} a^{3} + \frac{63229}{48194} a^{2} + \frac{13309}{24097} a - \frac{18735}{24097} \),  \( \frac{219}{48194} a^{19} + \frac{2440}{24097} a^{18} - \frac{2362}{24097} a^{17} + \frac{1393}{24097} a^{16} - \frac{2802}{24097} a^{15} - \frac{30137}{48194} a^{14} + \frac{11477}{24097} a^{13} - \frac{20327}{24097} a^{12} + \frac{20712}{24097} a^{11} + \frac{26845}{48194} a^{10} - \frac{1483}{48194} a^{9} + \frac{169523}{48194} a^{8} - \frac{71511}{24097} a^{7} + \frac{46315}{48194} a^{6} - \frac{23485}{24097} a^{5} - \frac{163851}{48194} a^{4} + \frac{121691}{48194} a^{3} - \frac{33767}{24097} a^{2} + \frac{16592}{24097} a + \frac{3865}{24097} \),  \( \frac{2440}{24097} a^{19} - \frac{5535}{48194} a^{18} + \frac{1393}{24097} a^{17} - \frac{2691}{24097} a^{16} - \frac{30137}{48194} a^{15} + \frac{16529}{48194} a^{14} - \frac{20327}{24097} a^{13} + \frac{24927}{48194} a^{12} + \frac{26845}{48194} a^{11} + \frac{5009}{24097} a^{10} + \frac{169523}{48194} a^{9} - \frac{66829}{48194} a^{8} + \frac{46315}{48194} a^{7} + \frac{65213}{48194} a^{6} - \frac{163851}{48194} a^{5} + \frac{45977}{48194} a^{4} - \frac{33767}{24097} a^{3} - \frac{39393}{48194} a^{2} + \frac{3865}{24097} a - \frac{52073}{48194} \),  \( \frac{14049}{24097} a^{19} + \frac{3898}{24097} a^{18} + \frac{23417}{24097} a^{17} + \frac{13497}{48194} a^{16} - \frac{41288}{24097} a^{15} - \frac{9960}{24097} a^{14} - \frac{124159}{24097} a^{13} - \frac{32888}{24097} a^{12} - \frac{175729}{48194} a^{11} - \frac{28183}{24097} a^{10} + \frac{236941}{24097} a^{9} + \frac{52937}{24097} a^{8} + \frac{347703}{48194} a^{7} + \frac{75669}{48194} a^{6} - \frac{214317}{48194} a^{5} - \frac{19806}{24097} a^{4} - \frac{218607}{48194} a^{3} - \frac{18806}{24097} a^{2} + \frac{45199}{48194} a + \frac{42569}{48194} \),  \( \frac{19519}{24097} a^{19} - \frac{6227}{48194} a^{18} + \frac{63153}{48194} a^{17} - \frac{11031}{48194} a^{16} - \frac{57144}{24097} a^{15} + \frac{5730}{24097} a^{14} - \frac{330173}{48194} a^{13} + \frac{50325}{48194} a^{12} - \frac{235539}{48194} a^{11} + \frac{32980}{24097} a^{10} + \frac{631417}{48194} a^{9} - \frac{29366}{24097} a^{8} + \frac{422991}{48194} a^{7} - \frac{34722}{24097} a^{6} - \frac{140460}{24097} a^{5} - \frac{55599}{48194} a^{4} - \frac{110045}{24097} a^{3} + \frac{8932}{24097} a^{2} + \frac{15364}{24097} a + \frac{26825}{48194} \),  \( \frac{3145}{48194} a^{19} + \frac{1458}{24097} a^{18} - \frac{2781}{24097} a^{17} + \frac{10711}{48194} a^{16} - \frac{13501}{24097} a^{15} + \frac{10217}{48194} a^{14} - \frac{780}{24097} a^{13} - \frac{12561}{24097} a^{12} + \frac{65075}{48194} a^{11} - \frac{83211}{48194} a^{10} + \frac{136929}{48194} a^{9} - \frac{63649}{48194} a^{8} - \frac{86859}{48194} a^{7} + \frac{14677}{24097} a^{6} - \frac{108189}{48194} a^{5} + \frac{124239}{48194} a^{4} - \frac{13787}{24097} a^{3} + \frac{14961}{24097} a^{2} - \frac{17387}{48194} a + \frac{34839}{48194} \),  \( \frac{3974}{24097} a^{19} - \frac{3898}{24097} a^{18} + \frac{8135}{24097} a^{17} - \frac{13497}{48194} a^{16} - \frac{9264}{24097} a^{15} + \frac{9960}{24097} a^{14} - \frac{37906}{24097} a^{13} + \frac{32888}{24097} a^{12} - \frac{75479}{48194} a^{11} + \frac{28183}{24097} a^{10} + \frac{51212}{24097} a^{9} - \frac{52937}{24097} a^{8} + \frac{136169}{48194} a^{7} - \frac{75669}{48194} a^{6} - \frac{19947}{48194} a^{5} + \frac{19806}{24097} a^{4} - \frac{9377}{48194} a^{3} + \frac{18806}{24097} a^{2} - \frac{50813}{48194} a - \frac{42569}{48194} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 228.08067214 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

20T992:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 1966080
The 280 conjugacy class representatives for t20n992 are not computed
Character table for t20n992 is not computed

Intermediate fields

5.5.38569.1, 10.2.1487567761.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.10.10.14$x^{10} + 5 x^{8} - 50 x^{6} - 58 x^{4} + 49 x^{2} + 21$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
38569Data not computed