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Properties

Label 20.0.29950357231...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{7}\cdot 13^{4}\cdot 41^{10}$
Root discriminant $18.79$
Ramified primes $5, 13, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T174

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 24, -57, 90, -111, 151, -210, 197, -67, -51, 42, 71, -145, 126, -95, 65, -29, 12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 12*x^18 - 29*x^17 + 65*x^16 - 95*x^15 + 126*x^14 - 145*x^13 + 71*x^12 + 42*x^11 - 51*x^10 - 67*x^9 + 197*x^8 - 210*x^7 + 151*x^6 - 111*x^5 + 90*x^4 - 57*x^3 + 24*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 12*x^18 - 29*x^17 + 65*x^16 - 95*x^15 + 126*x^14 - 145*x^13 + 71*x^12 + 42*x^11 - 51*x^10 - 67*x^9 + 197*x^8 - 210*x^7 + 151*x^6 - 111*x^5 + 90*x^4 - 57*x^3 + 24*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 12 x^{18} - 29 x^{17} + 65 x^{16} - 95 x^{15} + 126 x^{14} - 145 x^{13} + 71 x^{12} + 42 x^{11} - 51 x^{10} - 67 x^{9} + 197 x^{8} - 210 x^{7} + 151 x^{6} - 111 x^{5} + 90 x^{4} - 57 x^{3} + 24 x^{2} - 6 x + 1 \)

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:  \(29950357231036150387578125=5^{7}\cdot 13^{4}\cdot 41^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.79$
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:  $5, 13, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{507333271674884405} a^{19} - \frac{32553324757915637}{507333271674884405} a^{18} + \frac{50808785287849323}{507333271674884405} a^{17} - \frac{115696816594179918}{507333271674884405} a^{16} - \frac{204488427033677086}{507333271674884405} a^{15} + \frac{76109334572058838}{507333271674884405} a^{14} - \frac{34944023005455003}{507333271674884405} a^{13} + \frac{765441776329254}{507333271674884405} a^{12} + \frac{133863942351818864}{507333271674884405} a^{11} + \frac{43285719627067853}{101466654334976881} a^{10} - \frac{13660747288589721}{507333271674884405} a^{9} + \frac{220875557138248226}{507333271674884405} a^{8} - \frac{69493200814231171}{507333271674884405} a^{7} + \frac{96957650888963518}{507333271674884405} a^{6} + \frac{44365188709745882}{507333271674884405} a^{5} - \frac{41116842817059327}{507333271674884405} a^{4} + \frac{219520120154638866}{507333271674884405} a^{3} + \frac{48172345198981527}{101466654334976881} a^{2} - \frac{101981558438423861}{507333271674884405} a + \frac{66517904472945107}{507333271674884405}$

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:  \( -1 \) (order $2$)
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:  \( 38320.3608939 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T174:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 960
The 35 conjugacy class representatives for t20n174
Character table for t20n174 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.0.8405.1, 5.1.2665.1, 10.2.489492449225.1

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 24 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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.12.0.1$x^{12} + x^{2} - x + 2$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$41$41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$