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Properties

Label 20.8.39479125871...5849.1
Degree $20$
Signature $[8, 6]$
Discriminant $3^{10}\cdot 401^{8}$
Root discriminant $19.05$
Ramified primes $3, 401$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^4:D_5$ (as 20T85)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, 36, -3, -48, -125, 79, 245, -41, -157, 32, -125, 90, 158, -185, 29, 60, -49, 12, 6, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 6*x^18 + 12*x^17 - 49*x^16 + 60*x^15 + 29*x^14 - 185*x^13 + 158*x^12 + 90*x^11 - 125*x^10 + 32*x^9 - 157*x^8 - 41*x^7 + 245*x^6 + 79*x^5 - 125*x^4 - 48*x^3 - 3*x^2 + 36*x - 9)
 
gp: K = bnfinit(x^20 - 5*x^19 + 6*x^18 + 12*x^17 - 49*x^16 + 60*x^15 + 29*x^14 - 185*x^13 + 158*x^12 + 90*x^11 - 125*x^10 + 32*x^9 - 157*x^8 - 41*x^7 + 245*x^6 + 79*x^5 - 125*x^4 - 48*x^3 - 3*x^2 + 36*x - 9, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 6 x^{18} + 12 x^{17} - 49 x^{16} + 60 x^{15} + 29 x^{14} - 185 x^{13} + 158 x^{12} + 90 x^{11} - 125 x^{10} + 32 x^{9} - 157 x^{8} - 41 x^{7} + 245 x^{6} + 79 x^{5} - 125 x^{4} - 48 x^{3} - 3 x^{2} + 36 x - 9 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(39479125871598264344535849=3^{10}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.05$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13197} a^{18} - \frac{938}{13197} a^{17} + \frac{1}{4399} a^{16} + \frac{496}{4399} a^{15} - \frac{1936}{13197} a^{14} + \frac{1704}{4399} a^{13} + \frac{1757}{13197} a^{12} + \frac{3772}{13197} a^{11} + \frac{2087}{13197} a^{10} - \frac{582}{4399} a^{9} + \frac{4606}{13197} a^{8} + \frac{1136}{13197} a^{7} - \frac{6565}{13197} a^{6} + \frac{74}{159} a^{5} - \frac{3367}{13197} a^{4} + \frac{3592}{13197} a^{3} + \frac{1642}{13197} a^{2} - \frac{45}{4399} a - \frac{1893}{4399}$, $\frac{1}{1544649051898542651} a^{19} - \frac{6154880900239}{514883017299514217} a^{18} + \frac{45449717427347531}{1544649051898542651} a^{17} - \frac{167305921550507205}{514883017299514217} a^{16} + \frac{145189754157283358}{1544649051898542651} a^{15} - \frac{228783756776312192}{1544649051898542651} a^{14} + \frac{411377654311685081}{1544649051898542651} a^{13} + \frac{8162771304393104}{81297318520975929} a^{12} - \frac{149748244948577474}{1544649051898542651} a^{11} - \frac{190148531208805640}{1544649051898542651} a^{10} - \frac{605848008316206206}{1544649051898542651} a^{9} + \frac{246270438305628307}{1544649051898542651} a^{8} + \frac{1068539899477316}{514883017299514217} a^{7} - \frac{278782008932826379}{1544649051898542651} a^{6} - \frac{527390694825959042}{1544649051898542651} a^{5} + \frac{203487258979954109}{1544649051898542651} a^{4} + \frac{35411142495226938}{514883017299514217} a^{3} - \frac{302831796525397843}{1544649051898542651} a^{2} - \frac{42536920646155836}{514883017299514217} a + \frac{55125501074112150}{514883017299514217}$

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:  $13$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 159172.765399 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T85):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

5.5.160801.1, 10.4.698137963227.1, 10.8.698137963227.1, 10.6.2094413889681.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
401Data not computed