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Properties

Label 20.0.39375744125...4825.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{2}\cdot 7\cdot 43^{2}\cdot 151\cdot 36943^{2}$
Root discriminant $12.02$
Ramified primes $3, 5, 7, 43, 151, 36943$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1045

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 15, -21, 22, -4, -12, 25, -13, -16, 42, -46, 18, 21, -42, 34, -9, -9, 11, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 11*x^18 - 9*x^17 - 9*x^16 + 34*x^15 - 42*x^14 + 21*x^13 + 18*x^12 - 46*x^11 + 42*x^10 - 16*x^9 - 13*x^8 + 25*x^7 - 12*x^6 - 4*x^5 + 22*x^4 - 21*x^3 + 15*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 11*x^18 - 9*x^17 - 9*x^16 + 34*x^15 - 42*x^14 + 21*x^13 + 18*x^12 - 46*x^11 + 42*x^10 - 16*x^9 - 13*x^8 + 25*x^7 - 12*x^6 - 4*x^5 + 22*x^4 - 21*x^3 + 15*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 11 x^{18} - 9 x^{17} - 9 x^{16} + 34 x^{15} - 42 x^{14} + 21 x^{13} + 18 x^{12} - 46 x^{11} + 42 x^{10} - 16 x^{9} - 13 x^{8} + 25 x^{7} - 12 x^{6} - 4 x^{5} + 22 x^{4} - 21 x^{3} + 15 x^{2} - 6 x + 1 \)

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:  \(3937574412547571424825=3^{10}\cdot 5^{2}\cdot 7\cdot 43^{2}\cdot 151\cdot 36943^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.02$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 5, 7, 43, 151, 36943$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{167365427} a^{19} - \frac{59956051}{167365427} a^{18} + \frac{55787998}{167365427} a^{17} + \frac{75305089}{167365427} a^{16} - \frac{13444613}{167365427} a^{15} + \frac{47380738}{167365427} a^{14} - \frac{29476774}{167365427} a^{13} + \frac{26484414}{167365427} a^{12} + \frac{4703}{167365427} a^{11} + \frac{37460111}{167365427} a^{10} + \frac{41980009}{167365427} a^{9} - \frac{83602643}{167365427} a^{8} - \frac{673188}{167365427} a^{7} + \frac{11684780}{167365427} a^{6} + \frac{60045138}{167365427} a^{5} - \frac{59820715}{167365427} a^{4} + \frac{54978367}{167365427} a^{3} + \frac{15067877}{167365427} a^{2} - \frac{36345490}{167365427} a + \frac{35610853}{167365427}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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:  \( \frac{10078}{20939} a^{19} - \frac{44893}{20939} a^{18} + \frac{92781}{20939} a^{17} - \frac{71142}{20939} a^{16} - \frac{69361}{20939} a^{15} + \frac{280234}{20939} a^{14} - \frac{372829}{20939} a^{13} + \frac{219085}{20939} a^{12} + \frac{137511}{20939} a^{11} - \frac{408045}{20939} a^{10} + \frac{402460}{20939} a^{9} - \frac{185134}{20939} a^{8} - \frac{131725}{20939} a^{7} + \frac{245106}{20939} a^{6} - \frac{105336}{20939} a^{5} - \frac{11389}{20939} a^{4} + \frac{245778}{20939} a^{3} - \frac{170174}{20939} a^{2} + \frac{161885}{20939} a - \frac{33500}{20939} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 734.197779319 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1045:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 29491200
The 702 conjugacy class representatives for t20n1045 are not computed
Character table for t20n1045 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.0.386017407.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.8.0.1}{8} }$ R R R $16{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
3Data not computed
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.12.0.1$x^{12} - x^{3} - 2 x + 3$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$
7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
7.8.0.1$x^{8} - x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.6.0.1$x^{6} - x + 26$$1$$6$$0$$C_6$$[\ ]^{6}$
43.8.0.1$x^{8} - 3 x + 18$$1$$8$$0$$C_8$$[\ ]^{8}$
$151$151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
151.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
151.6.0.1$x^{6} - x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$
36943Data not computed