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Properties

Label 20.8.23533415996...8125.1
Degree $20$
Signature $[8, 6]$
Discriminant $3^{8}\cdot 5^{10}\cdot 239^{8}\cdot 34501$
Root discriminant $52.31$
Ramified primes $3, 5, 239, 34501$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T525

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, 12, -227, 611, -584, -348, 1816, -2457, 1389, 705, -2252, 2273, -1174, -15, 615, -624, 384, -165, 50, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 50*x^18 - 165*x^17 + 384*x^16 - 624*x^15 + 615*x^14 - 15*x^13 - 1174*x^12 + 2273*x^11 - 2252*x^10 + 705*x^9 + 1389*x^8 - 2457*x^7 + 1816*x^6 - 348*x^5 - 584*x^4 + 611*x^3 - 227*x^2 + 12*x + 19)
 
gp: K = bnfinit(x^20 - 10*x^19 + 50*x^18 - 165*x^17 + 384*x^16 - 624*x^15 + 615*x^14 - 15*x^13 - 1174*x^12 + 2273*x^11 - 2252*x^10 + 705*x^9 + 1389*x^8 - 2457*x^7 + 1816*x^6 - 348*x^5 - 584*x^4 + 611*x^3 - 227*x^2 + 12*x + 19, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 50 x^{18} - 165 x^{17} + 384 x^{16} - 624 x^{15} + 615 x^{14} - 15 x^{13} - 1174 x^{12} + 2273 x^{11} - 2252 x^{10} + 705 x^{9} + 1389 x^{8} - 2457 x^{7} + 1816 x^{6} - 348 x^{5} - 584 x^{4} + 611 x^{3} - 227 x^{2} + 12 x + 19 \)

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:  \(23533415996900471975491413486328125=3^{8}\cdot 5^{10}\cdot 239^{8}\cdot 34501\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.31$
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:  $3, 5, 239, 34501$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{11} a^{16} + \frac{3}{11} a^{15} + \frac{5}{11} a^{14} - \frac{5}{11} a^{13} + \frac{2}{11} a^{12} - \frac{3}{11} a^{11} + \frac{1}{11} a^{8} - \frac{4}{11} a^{7} - \frac{1}{11} a^{6} + \frac{5}{11} a^{4} + \frac{1}{11} a^{3} + \frac{4}{11} a^{2} + \frac{2}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{17} - \frac{4}{11} a^{15} + \frac{2}{11} a^{14} - \frac{5}{11} a^{13} + \frac{2}{11} a^{12} - \frac{2}{11} a^{11} + \frac{1}{11} a^{9} + \frac{4}{11} a^{8} + \frac{3}{11} a^{6} + \frac{5}{11} a^{5} - \frac{3}{11} a^{4} + \frac{1}{11} a^{3} + \frac{1}{11} a^{2} + \frac{4}{11} a + \frac{3}{11}$, $\frac{1}{5907} a^{18} - \frac{3}{1969} a^{17} - \frac{45}{1969} a^{16} + \frac{428}{1969} a^{15} - \frac{116}{1969} a^{14} + \frac{960}{1969} a^{13} + \frac{280}{1969} a^{12} + \frac{291}{1969} a^{11} + \frac{1145}{5907} a^{10} - \frac{2018}{5907} a^{9} + \frac{955}{5907} a^{8} + \frac{2408}{5907} a^{7} - \frac{345}{1969} a^{6} + \frac{689}{5907} a^{5} - \frac{67}{179} a^{4} + \frac{224}{5907} a^{3} - \frac{4}{1969} a^{2} + \frac{376}{5907} a - \frac{424}{5907}$, $\frac{1}{112233} a^{19} - \frac{72}{37411} a^{17} + \frac{23}{37411} a^{16} - \frac{8078}{37411} a^{15} + \frac{5823}{37411} a^{14} - \frac{12739}{37411} a^{13} + \frac{977}{1969} a^{12} - \frac{14626}{112233} a^{11} + \frac{31915}{112233} a^{10} + \frac{12328}{112233} a^{9} - \frac{12625}{112233} a^{8} + \frac{2941}{37411} a^{7} + \frac{26816}{112233} a^{6} + \frac{15113}{37411} a^{5} + \frac{45302}{112233} a^{4} - \frac{17053}{37411} a^{3} + \frac{41617}{112233} a^{2} - \frac{50203}{112233} a - \frac{896}{1969}$

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:  $13$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 13962473062.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 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 $20$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
239Data not computed
34501Data not computed