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Properties

Label 20.4.10029834359...6689.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{10}\cdot 47^{8}\cdot 61^{10}$
Root discriminant $63.11$
Ramified primes $3, 47, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_5\times A_4$ (as 20T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -23, -180, -803, -1584, 1068, 840, -5805, 11955, -11221, -4020, 5107, -1728, 1011, -63, 0, -3, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^17 - 63*x^15 + 1011*x^14 - 1728*x^13 + 5107*x^12 - 4020*x^11 - 11221*x^10 + 11955*x^9 - 5805*x^8 + 840*x^7 + 1068*x^6 - 1584*x^5 - 803*x^4 - 180*x^3 - 23*x^2 + 1)
 
gp: K = bnfinit(x^20 - 3*x^17 - 63*x^15 + 1011*x^14 - 1728*x^13 + 5107*x^12 - 4020*x^11 - 11221*x^10 + 11955*x^9 - 5805*x^8 + 840*x^7 + 1068*x^6 - 1584*x^5 - 803*x^4 - 180*x^3 - 23*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{17} - 63 x^{15} + 1011 x^{14} - 1728 x^{13} + 5107 x^{12} - 4020 x^{11} - 11221 x^{10} + 11955 x^{9} - 5805 x^{8} + 840 x^{7} + 1068 x^{6} - 1584 x^{5} - 803 x^{4} - 180 x^{3} - 23 x^{2} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1002983435921998221381095607508396689=3^{10}\cdot 47^{8}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.11$
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:  $3, 47, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} + \frac{4}{23} a^{17} + \frac{4}{23} a^{16} + \frac{11}{23} a^{15} - \frac{4}{23} a^{14} - \frac{4}{23} a^{13} + \frac{8}{23} a^{12} + \frac{8}{23} a^{11} + \frac{6}{23} a^{10} + \frac{2}{23} a^{9} + \frac{8}{23} a^{8} + \frac{3}{23} a^{7} - \frac{1}{23} a^{6} - \frac{5}{23} a^{5} + \frac{2}{23} a^{4} + \frac{2}{23} a^{3} + \frac{9}{23} a^{2} - \frac{7}{23} a + \frac{2}{23}$, $\frac{1}{47436463475610304004705221682195540303} a^{19} + \frac{864366485789711997064621432571469242}{47436463475610304004705221682195540303} a^{18} + \frac{13524533700658985266942549115297246202}{47436463475610304004705221682195540303} a^{17} + \frac{10381002064163479193489852124746525243}{47436463475610304004705221682195540303} a^{16} - \frac{3392603301368280855160396759553857151}{47436463475610304004705221682195540303} a^{15} + \frac{16947523984443880462959580271462128868}{47436463475610304004705221682195540303} a^{14} - \frac{3867608108735988955027409220110675735}{47436463475610304004705221682195540303} a^{13} - \frac{14266558282454226121236314784419644741}{47436463475610304004705221682195540303} a^{12} + \frac{19378595296742178145525286321061703065}{47436463475610304004705221682195540303} a^{11} + \frac{9856493166272408794635725944152654834}{47436463475610304004705221682195540303} a^{10} + \frac{6174137084610467675407909753782927339}{47436463475610304004705221682195540303} a^{9} - \frac{16332440920437692235613845149863488709}{47436463475610304004705221682195540303} a^{8} + \frac{11463810322766812209697681561740226950}{47436463475610304004705221682195540303} a^{7} + \frac{15982873258196072909513056722125871888}{47436463475610304004705221682195540303} a^{6} + \frac{2938003194851982681330752782325638944}{47436463475610304004705221682195540303} a^{5} - \frac{17917563505740507337483881096856643618}{47436463475610304004705221682195540303} a^{4} + \frac{13680270000373838704193112761979340081}{47436463475610304004705221682195540303} a^{3} - \frac{15179321565651082378909011144145635583}{47436463475610304004705221682195540303} a^{2} + \frac{21309755516483080942135966957373972453}{47436463475610304004705221682195540303} a - \frac{5724057706598298206810875047512540183}{47436463475610304004705221682195540303}$

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:  $11$
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:  \( 13453127150.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\times A_4$ (as 20T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 16 conjugacy class representatives for $D_5\times A_4$
Character table for $D_5\times A_4$

Intermediate fields

4.4.33489.1, 5.1.2209.1

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

Sibling fields

Degree 30 siblings: data not computed
Degree 40 sibling: 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 $15{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $15{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $15{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ R ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $15{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
3Data not computed
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
61Data not computed