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Properties

Label 20.16.3291332223...5625.1
Degree $20$
Signature $[16, 2]$
Discriminant $5^{10}\cdot 71^{2}\cdot 401^{8}$
Root discriminant $37.66$
Ramified primes $5, 71, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T141

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 30, -181, -438, 1254, 2227, -3903, -5702, 6400, 8079, -6059, -6400, 3556, 2743, -1341, -593, 295, 54, -31, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 31*x^18 + 54*x^17 + 295*x^16 - 593*x^15 - 1341*x^14 + 2743*x^13 + 3556*x^12 - 6400*x^11 - 6059*x^10 + 8079*x^9 + 6400*x^8 - 5702*x^7 - 3903*x^6 + 2227*x^5 + 1254*x^4 - 438*x^3 - 181*x^2 + 30*x + 9)
 
gp: K = bnfinit(x^20 - x^19 - 31*x^18 + 54*x^17 + 295*x^16 - 593*x^15 - 1341*x^14 + 2743*x^13 + 3556*x^12 - 6400*x^11 - 6059*x^10 + 8079*x^9 + 6400*x^8 - 5702*x^7 - 3903*x^6 + 2227*x^5 + 1254*x^4 - 438*x^3 - 181*x^2 + 30*x + 9, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 31 x^{18} + 54 x^{17} + 295 x^{16} - 593 x^{15} - 1341 x^{14} + 2743 x^{13} + 3556 x^{12} - 6400 x^{11} - 6059 x^{10} + 8079 x^{9} + 6400 x^{8} - 5702 x^{7} - 3903 x^{6} + 2227 x^{5} + 1254 x^{4} - 438 x^{3} - 181 x^{2} + 30 x + 9 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(32913322237994155701330478515625=5^{10}\cdot 71^{2}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.66$
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:  $5, 71, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} - \frac{7}{23} a^{17} + \frac{1}{23} a^{16} + \frac{3}{23} a^{15} - \frac{9}{23} a^{14} + \frac{6}{23} a^{13} + \frac{1}{23} a^{12} + \frac{9}{23} a^{11} - \frac{4}{23} a^{10} - \frac{3}{23} a^{9} + \frac{2}{23} a^{8} + \frac{1}{23} a^{7} + \frac{3}{23} a^{6} - \frac{3}{23} a^{5} - \frac{5}{23} a^{4} + \frac{10}{23} a^{3} + \frac{2}{23} a^{2} + \frac{2}{23} a - \frac{6}{23}$, $\frac{1}{163900457060182889631} a^{19} + \frac{2253673001067039164}{163900457060182889631} a^{18} - \frac{7016403115981377598}{163900457060182889631} a^{17} - \frac{20679482205192948287}{54633485686727629877} a^{16} - \frac{49244738293817939810}{163900457060182889631} a^{15} + \frac{4685418887121136072}{163900457060182889631} a^{14} - \frac{4391682646919850262}{54633485686727629877} a^{13} + \frac{3032965574702565923}{7126106828703603897} a^{12} + \frac{32811295129189813861}{163900457060182889631} a^{11} + \frac{77306250258024203363}{163900457060182889631} a^{10} + \frac{28346799539637779599}{163900457060182889631} a^{9} - \frac{6035589333551108853}{54633485686727629877} a^{8} + \frac{14229521466838236766}{163900457060182889631} a^{7} - \frac{55308737573591140871}{163900457060182889631} a^{6} - \frac{24270578049108651493}{54633485686727629877} a^{5} + \frac{3092580550259280472}{12607727466167914587} a^{4} - \frac{23707137993430693423}{54633485686727629877} a^{3} + \frac{11725466485492454412}{54633485686727629877} a^{2} - \frac{61285077287366226565}{163900457060182889631} a + \frac{27131759973040690005}{54633485686727629877}$

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:  $17$
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:  \( 834605177.945 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T141:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n141
Character table for t20n141 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.160801.1, 10.8.5737013355221875.1, 10.10.80803005003125.1, 10.8.1835844273671.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$71$71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
401Data not computed