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Properties

Label 20.10.1467214799...9375.4
Degree $20$
Signature $[10, 5]$
Discriminant $-\,3^{10}\cdot 5^{10}\cdot 239^{9}$
Root discriminant $45.53$
Ramified primes $3, 5, 239$
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![55, -320, -801, 7323, -9342, -7773, 16903, 375, -10683, 3283, 2486, -2150, 592, 263, -356, 148, -29, -6, 12, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 12*x^18 - 6*x^17 - 29*x^16 + 148*x^15 - 356*x^14 + 263*x^13 + 592*x^12 - 2150*x^11 + 2486*x^10 + 3283*x^9 - 10683*x^8 + 375*x^7 + 16903*x^6 - 7773*x^5 - 9342*x^4 + 7323*x^3 - 801*x^2 - 320*x + 55)
 
gp: K = bnfinit(x^20 - 6*x^19 + 12*x^18 - 6*x^17 - 29*x^16 + 148*x^15 - 356*x^14 + 263*x^13 + 592*x^12 - 2150*x^11 + 2486*x^10 + 3283*x^9 - 10683*x^8 + 375*x^7 + 16903*x^6 - 7773*x^5 - 9342*x^4 + 7323*x^3 - 801*x^2 - 320*x + 55, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 12 x^{18} - 6 x^{17} - 29 x^{16} + 148 x^{15} - 356 x^{14} + 263 x^{13} + 592 x^{12} - 2150 x^{11} + 2486 x^{10} + 3283 x^{9} - 10683 x^{8} + 375 x^{7} + 16903 x^{6} - 7773 x^{5} - 9342 x^{4} + 7323 x^{3} - 801 x^{2} - 320 x + 55 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1467214799841538367562738193359375=-\,3^{10}\cdot 5^{10}\cdot 239^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{237} a^{18} + \frac{26}{237} a^{17} - \frac{97}{237} a^{16} - \frac{28}{79} a^{15} - \frac{26}{79} a^{14} - \frac{92}{237} a^{13} - \frac{18}{79} a^{12} + \frac{8}{79} a^{11} + \frac{34}{237} a^{10} + \frac{18}{79} a^{9} - \frac{17}{79} a^{8} - \frac{104}{237} a^{7} + \frac{89}{237} a^{6} - \frac{112}{237} a^{5} - \frac{41}{237} a^{4} + \frac{85}{237} a^{3} - \frac{12}{79} a^{2} - \frac{107}{237} a + \frac{26}{237}$, $\frac{1}{263653422622844769133580674212225} a^{19} - \frac{5783464606582576577358891622}{52730684524568953826716134842445} a^{18} - \frac{42803217947635948291425243014566}{87884474207614923044526891404075} a^{17} - \frac{10806970100350171529553358327139}{263653422622844769133580674212225} a^{16} - \frac{24193133494736169461264861881616}{87884474207614923044526891404075} a^{15} - \frac{7093821633223435743636767878297}{52730684524568953826716134842445} a^{14} - \frac{124152088663508981683200553976941}{263653422622844769133580674212225} a^{13} - \frac{41225372397143078871987006083916}{87884474207614923044526891404075} a^{12} - \frac{11269350211886155176746939989616}{263653422622844769133580674212225} a^{11} + \frac{49189450175817344080762676390864}{263653422622844769133580674212225} a^{10} - \frac{8013073497615851293500351781823}{17576894841522984608905378280815} a^{9} + \frac{93217867896075710781573803074063}{263653422622844769133580674212225} a^{8} - \frac{22303438602536849107810677848632}{52730684524568953826716134842445} a^{7} - \frac{2508538124628783192289038481549}{17576894841522984608905378280815} a^{6} - \frac{63730766527718612668145753727157}{263653422622844769133580674212225} a^{5} - \frac{8068638208276384646292799066028}{17576894841522984608905378280815} a^{4} + \frac{126669714626710542864926942084288}{263653422622844769133580674212225} a^{3} - \frac{2977068346002459932650651999079}{263653422622844769133580674212225} a^{2} - \frac{775222943875639611355558250108}{1818299466364446683679866718705} a + \frac{1071141799894082101842652037014}{4793698593142632166065103167495}$

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:  $14$
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:  \( 3791542474.65 \) (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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$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}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed