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Properties

Label 20.12.1948135206...8125.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{8}\cdot 5^{11}\cdot 239^{10}$
Root discriminant $58.14$
Ramified primes $3, 5, 239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, -108, 177, -1983, 9986, -18569, -17734, 106289, -57180, -119589, 105936, 25779, -42307, 3150, 6410, -1739, -232, 194, -27, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 27*x^18 + 194*x^17 - 232*x^16 - 1739*x^15 + 6410*x^14 + 3150*x^13 - 42307*x^12 + 25779*x^11 + 105936*x^10 - 119589*x^9 - 57180*x^8 + 106289*x^7 - 17734*x^6 - 18569*x^5 + 9986*x^4 - 1983*x^3 + 177*x^2 - 108*x - 9)
 
gp: K = bnfinit(x^20 - 4*x^19 - 27*x^18 + 194*x^17 - 232*x^16 - 1739*x^15 + 6410*x^14 + 3150*x^13 - 42307*x^12 + 25779*x^11 + 105936*x^10 - 119589*x^9 - 57180*x^8 + 106289*x^7 - 17734*x^6 - 18569*x^5 + 9986*x^4 - 1983*x^3 + 177*x^2 - 108*x - 9, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 27 x^{18} + 194 x^{17} - 232 x^{16} - 1739 x^{15} + 6410 x^{14} + 3150 x^{13} - 42307 x^{12} + 25779 x^{11} + 105936 x^{10} - 119589 x^{9} - 57180 x^{8} + 106289 x^{7} - 17734 x^{6} - 18569 x^{5} + 9986 x^{4} - 1983 x^{3} + 177 x^{2} - 108 x - 9 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(194813520645626483248608015673828125=3^{8}\cdot 5^{11}\cdot 239^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.14$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{15} - \frac{4}{9} a^{11} - \frac{1}{3} a^{10} + \frac{2}{9} a^{9} + \frac{4}{9} a^{8} - \frac{1}{3} a^{7} - \frac{1}{9} a^{6} + \frac{1}{3} a^{5} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{12} - \frac{4}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{8} - \frac{1}{9} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{4}{9} a^{4} + \frac{4}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{13} - \frac{4}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{3} a^{9} - \frac{1}{9} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{4}{9} a^{5} + \frac{4}{9} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{621} a^{18} - \frac{4}{621} a^{17} - \frac{4}{207} a^{16} - \frac{4}{621} a^{15} - \frac{67}{621} a^{14} - \frac{5}{621} a^{13} + \frac{1}{27} a^{12} - \frac{34}{69} a^{11} - \frac{286}{621} a^{10} + \frac{94}{207} a^{9} - \frac{89}{207} a^{8} - \frac{89}{207} a^{7} - \frac{17}{207} a^{6} + \frac{302}{621} a^{5} - \frac{214}{621} a^{4} + \frac{175}{621} a^{3} - \frac{124}{621} a^{2} - \frac{62}{207} a + \frac{37}{207}$, $\frac{1}{269737882183756087393338606615212569239} a^{19} + \frac{48914055791892360280178248039310954}{89912627394585362464446202205070856413} a^{18} + \frac{3318992726838307993126504972702686200}{269737882183756087393338606615212569239} a^{17} - \frac{5845148918027084113457892585911454403}{269737882183756087393338606615212569239} a^{16} + \frac{6469936825718509478413984840575359662}{269737882183756087393338606615212569239} a^{15} - \frac{1525635378752122006596446158315479427}{29970875798195120821482067401690285471} a^{14} + \frac{137991529033777647655979565038122706}{1341979513352020335290241823956281439} a^{13} + \frac{22641813451352126757154013734889688524}{269737882183756087393338606615212569239} a^{12} + \frac{30881042302549964584539534604487106086}{269737882183756087393338606615212569239} a^{11} + \frac{72387416066493095528941983921368008961}{269737882183756087393338606615212569239} a^{10} + \frac{28936378209647616372280631061712008596}{89912627394585362464446202205070856413} a^{9} + \frac{1290813594067785930011364062242033982}{9990291932731706940494022467230095157} a^{8} - \frac{852734891669064124553903405583054}{212559402824078871074340903558087131} a^{7} + \frac{84681603057237604301226667549729645784}{269737882183756087393338606615212569239} a^{6} - \frac{85842879905356450874111581219428515303}{269737882183756087393338606615212569239} a^{5} + \frac{31454758793426941746401098493398296118}{89912627394585362464446202205070856413} a^{4} + \frac{24807835010274493971915608551035818312}{89912627394585362464446202205070856413} a^{3} - \frac{31181818633893179086995442004540872102}{269737882183756087393338606615212569239} a^{2} - \frac{18564487197183739102464353483700786541}{89912627394585362464446202205070856413} a + \frac{10982772608419519285886720514381554599}{89912627394585362464446202205070856413}$

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:  $15$
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:  \( 53254684647.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 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 $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$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