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Properties

Label 20.16.3410541143...8125.1
Degree $20$
Signature $[16, 2]$
Discriminant $3^{8}\cdot 5^{11}\cdot 239^{8}$
Root discriminant $33.62$
Ramified primes $3, 5, 239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T144

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![95, -270, -1110, 4520, -756, -12858, 10965, 10557, -13628, -2465, 6985, -1533, -1843, 1492, 278, -531, -4, 90, -8, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 8*x^18 + 90*x^17 - 4*x^16 - 531*x^15 + 278*x^14 + 1492*x^13 - 1843*x^12 - 1533*x^11 + 6985*x^10 - 2465*x^9 - 13628*x^8 + 10557*x^7 + 10965*x^6 - 12858*x^5 - 756*x^4 + 4520*x^3 - 1110*x^2 - 270*x + 95)
 
gp: K = bnfinit(x^20 - 6*x^19 - 8*x^18 + 90*x^17 - 4*x^16 - 531*x^15 + 278*x^14 + 1492*x^13 - 1843*x^12 - 1533*x^11 + 6985*x^10 - 2465*x^9 - 13628*x^8 + 10557*x^7 + 10965*x^6 - 12858*x^5 - 756*x^4 + 4520*x^3 - 1110*x^2 - 270*x + 95, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 8 x^{18} + 90 x^{17} - 4 x^{16} - 531 x^{15} + 278 x^{14} + 1492 x^{13} - 1843 x^{12} - 1533 x^{11} + 6985 x^{10} - 2465 x^{9} - 13628 x^{8} + 10557 x^{7} + 10965 x^{6} - 12858 x^{5} - 756 x^{4} + 4520 x^{3} - 1110 x^{2} - 270 x + 95 \)

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:  \(3410541143285770263976611328125=3^{8}\cdot 5^{11}\cdot 239^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.62$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11} a^{18} - \frac{3}{11} a^{17} - \frac{1}{11} a^{16} - \frac{5}{11} a^{15} - \frac{2}{11} a^{14} - \frac{1}{11} a^{13} + \frac{1}{11} a^{12} + \frac{5}{11} a^{11} + \frac{3}{11} a^{10} - \frac{3}{11} a^{9} - \frac{5}{11} a^{8} + \frac{2}{11} a^{7} + \frac{4}{11} a^{6} - \frac{3}{11} a^{5} - \frac{2}{11} a^{4} + \frac{2}{11} a^{3} - \frac{1}{11} a^{2} - \frac{5}{11} a + \frac{3}{11}$, $\frac{1}{5643661110765479223381751} a^{19} - \frac{100284863020751540847757}{5643661110765479223381751} a^{18} - \frac{719998226677019572769484}{5643661110765479223381751} a^{17} + \frac{92615533493443613074281}{513060100978679929398341} a^{16} + \frac{1898497023854040579513853}{5643661110765479223381751} a^{15} - \frac{2158305722540760919659920}{5643661110765479223381751} a^{14} + \frac{1090853717204011819016943}{5643661110765479223381751} a^{13} - \frac{34671867364237064918177}{513060100978679929398341} a^{12} - \frac{159806540286354770284187}{513060100978679929398341} a^{11} - \frac{1798514900428706117919818}{5643661110765479223381751} a^{10} - \frac{1036199251240930411108299}{5643661110765479223381751} a^{9} - \frac{129382347203753274298259}{297034795303446274914829} a^{8} - \frac{788300378351589688339112}{5643661110765479223381751} a^{7} - \frac{570774507259737742213671}{5643661110765479223381751} a^{6} - \frac{2230872866803814112467058}{5643661110765479223381751} a^{5} + \frac{1662706774131965670134016}{5643661110765479223381751} a^{4} + \frac{27065383011906116613069}{513060100978679929398341} a^{3} + \frac{209938236283955197161786}{513060100978679929398341} a^{2} - \frac{1751381070710909764776303}{5643661110765479223381751} a + \frac{24648896648468011683065}{297034795303446274914829}$

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:  \( 324130870.888 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T144:

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 t20n144
Character table for t20n144 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 $20$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed