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Properties

Label 20.8.36019579053...0625.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{10}\cdot 19^{6}\cdot 97^{2}\cdot 1699^{4}$
Root discriminant $37.83$
Ramified primes $5, 19, 97, 1699$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T760

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-29, 182, 1519, -17141, 48367, -3008, -113443, 161032, -103689, 29988, 2627, -9483, 7041, -1811, -396, 478, -194, 16, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 6*x^18 + 16*x^17 - 194*x^16 + 478*x^15 - 396*x^14 - 1811*x^13 + 7041*x^12 - 9483*x^11 + 2627*x^10 + 29988*x^9 - 103689*x^8 + 161032*x^7 - 113443*x^6 - 3008*x^5 + 48367*x^4 - 17141*x^3 + 1519*x^2 + 182*x - 29)
 
gp: K = bnfinit(x^20 - 4*x^19 + 6*x^18 + 16*x^17 - 194*x^16 + 478*x^15 - 396*x^14 - 1811*x^13 + 7041*x^12 - 9483*x^11 + 2627*x^10 + 29988*x^9 - 103689*x^8 + 161032*x^7 - 113443*x^6 - 3008*x^5 + 48367*x^4 - 17141*x^3 + 1519*x^2 + 182*x - 29, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 6 x^{18} + 16 x^{17} - 194 x^{16} + 478 x^{15} - 396 x^{14} - 1811 x^{13} + 7041 x^{12} - 9483 x^{11} + 2627 x^{10} + 29988 x^{9} - 103689 x^{8} + 161032 x^{7} - 113443 x^{6} - 3008 x^{5} + 48367 x^{4} - 17141 x^{3} + 1519 x^{2} + 182 x - 29 \)

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:  \(36019579053467556544112587890625=5^{10}\cdot 19^{6}\cdot 97^{2}\cdot 1699^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.83$
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:  $5, 19, 97, 1699$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{144388068858982794975745192503119043407359170491} a^{19} + \frac{41631421771054807769092750912260589531729317631}{144388068858982794975745192503119043407359170491} a^{18} - \frac{19845416765879123841142908378002396808703885588}{144388068858982794975745192503119043407359170491} a^{17} + \frac{4496863531347345944480425756367435924801140262}{144388068858982794975745192503119043407359170491} a^{16} + \frac{21041050101974148011393386083505222294592201457}{144388068858982794975745192503119043407359170491} a^{15} - \frac{1136781728196809391281584566838770396207280707}{3357862066487971976180120755886489381566492337} a^{14} - \frac{17167875089515795757139143941762151459961973056}{144388068858982794975745192503119043407359170491} a^{13} - \frac{58147146370719747096899906909231007711086218339}{144388068858982794975745192503119043407359170491} a^{12} + \frac{723578880277171635395806794350989280422691003}{3357862066487971976180120755886489381566492337} a^{11} + \frac{18676302471173418050855521930251261823771090711}{144388068858982794975745192503119043407359170491} a^{10} + \frac{69898298797098197017123035054516825594892160542}{144388068858982794975745192503119043407359170491} a^{9} + \frac{28559268764945223297763555351050385790231436422}{144388068858982794975745192503119043407359170491} a^{8} - \frac{47279788870806270875829929707566718325657407738}{144388068858982794975745192503119043407359170491} a^{7} - \frac{51884142408374091672737751710660173453594838552}{144388068858982794975745192503119043407359170491} a^{6} - \frac{917108497764496813351524152537144990739820360}{3357862066487971976180120755886489381566492337} a^{5} + \frac{47981851208461312415875538356224492098512226608}{144388068858982794975745192503119043407359170491} a^{4} - \frac{40761530265248197103388767579895188834532227861}{144388068858982794975745192503119043407359170491} a^{3} + \frac{57538792160460243229755859894961415758752833206}{144388068858982794975745192503119043407359170491} a^{2} - \frac{67346489392601410217310221866111003224224456661}{144388068858982794975745192503119043407359170491} a + \frac{53893341354679529675593463945446291789857642092}{144388068858982794975745192503119043407359170491}$

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:  \( 176414172.77 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T760:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n760 are not computed
Character table for t20n760 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.3256446753125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$97$97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
1699Data not computed