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Properties

Label 20.0.14854276432...7797.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 13^{5}\cdot 19^{4}\cdot 151^{4}$
Root discriminant $16.17$
Ramified primes $3, 13, 19, 151$
Class number $1$
Class group Trivial
Galois group 20T174

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -1, -1, 1, 4, -5, 0, 1, 0, 14, -21, 9, 0, 0, -4, 1, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + x^16 - 4*x^15 + 9*x^12 - 21*x^11 + 14*x^10 + x^8 - 5*x^6 + 4*x^5 + x^4 - x^3 - x^2 + x + 1)
 
gp: K = bnfinit(x^20 + x^16 - 4*x^15 + 9*x^12 - 21*x^11 + 14*x^10 + x^8 - 5*x^6 + 4*x^5 + x^4 - x^3 - x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{20} + x^{16} - 4 x^{15} + 9 x^{12} - 21 x^{11} + 14 x^{10} + x^{8} - 5 x^{6} + 4 x^{5} + x^{4} - x^{3} - x^{2} + x + 1 \)

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:  \(1485427643260203878707797=3^{10}\cdot 13^{5}\cdot 19^{4}\cdot 151^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.17$
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:  $3, 13, 19, 151$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17} a^{18} - \frac{1}{17} a^{17} - \frac{2}{17} a^{16} + \frac{5}{17} a^{15} + \frac{2}{17} a^{14} - \frac{4}{17} a^{13} - \frac{2}{17} a^{12} - \frac{3}{17} a^{11} + \frac{1}{17} a^{10} + \frac{4}{17} a^{9} + \frac{7}{17} a^{8} - \frac{2}{17} a^{7} - \frac{1}{17} a^{6} + \frac{7}{17} a^{5} + \frac{8}{17} a^{4} - \frac{8}{17} a^{3} + \frac{2}{17} a^{2} + \frac{4}{17} a + \frac{6}{17}$, $\frac{1}{23681} a^{19} - \frac{667}{23681} a^{18} - \frac{7836}{23681} a^{17} - \frac{4103}{23681} a^{16} - \frac{4722}{23681} a^{15} + \frac{9748}{23681} a^{14} + \frac{4787}{23681} a^{13} - \frac{8531}{23681} a^{12} - \frac{11363}{23681} a^{11} + \frac{9538}{23681} a^{10} + \frac{5571}{23681} a^{9} - \frac{9084}{23681} a^{8} + \frac{872}{23681} a^{7} - \frac{7708}{23681} a^{6} + \frac{5240}{23681} a^{5} - \frac{9790}{23681} a^{4} - \frac{4632}{23681} a^{3} + \frac{9620}{23681} a^{2} - \frac{4562}{23681} a + \frac{543}{23681}$

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

Class group and class number

Trivial group, which has 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:  \( -\frac{719}{17} a^{19} + \frac{384}{17} a^{18} - \frac{335}{17} a^{17} + \frac{118}{17} a^{16} - \frac{903}{17} a^{15} + \frac{3260}{17} a^{14} - \frac{1999}{17} a^{13} + \frac{1399}{17} a^{12} - \frac{7245}{17} a^{11} + \frac{19195}{17} a^{10} - \frac{21401}{17} a^{9} + \frac{13798}{17} a^{8} - \frac{9576}{17} a^{7} + \frac{6114}{17} a^{6} - \frac{396}{17} a^{5} - \frac{2334}{17} a^{4} + \frac{834}{17} a^{3} + \frac{160}{17} a^{2} + \frac{670}{17} a - \frac{1007}{17} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 16956.6944135 \)
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

\(\Q(\sqrt{-3}) \), 4.0.117.1, 5.1.2869.1, 10.0.2000172123.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ R $20$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ $20$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$13$13.10.5.2$x^{10} - 57122 x^{2} + 2227758$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
13.10.0.1$x^{10} + 2 x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
$151$$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
151.2.0.1$x^{2} - x + 12$$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}$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$
151.4.2.1$x^{4} + 3473 x^{2} + 3283344$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$