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Properties

Label 20.4.77541908691...0625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{8}\cdot 19^{8}\cdot 43^{8}$
Root discriminant $27.83$
Ramified primes $5, 19, 43$
Class number $1$
Class group Trivial
Galois group $C_2^4:D_5$ (as 20T39)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43, -110, 257, -763, 1194, 778, -5662, 10409, -11806, 9584, -6043, 3107, -1118, -43, 531, -548, 349, -156, 49, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 49*x^18 - 156*x^17 + 349*x^16 - 548*x^15 + 531*x^14 - 43*x^13 - 1118*x^12 + 3107*x^11 - 6043*x^10 + 9584*x^9 - 11806*x^8 + 10409*x^7 - 5662*x^6 + 778*x^5 + 1194*x^4 - 763*x^3 + 257*x^2 - 110*x + 43)
 
gp: K = bnfinit(x^20 - 10*x^19 + 49*x^18 - 156*x^17 + 349*x^16 - 548*x^15 + 531*x^14 - 43*x^13 - 1118*x^12 + 3107*x^11 - 6043*x^10 + 9584*x^9 - 11806*x^8 + 10409*x^7 - 5662*x^6 + 778*x^5 + 1194*x^4 - 763*x^3 + 257*x^2 - 110*x + 43, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 49 x^{18} - 156 x^{17} + 349 x^{16} - 548 x^{15} + 531 x^{14} - 43 x^{13} - 1118 x^{12} + 3107 x^{11} - 6043 x^{10} + 9584 x^{9} - 11806 x^{8} + 10409 x^{7} - 5662 x^{6} + 778 x^{5} + 1194 x^{4} - 763 x^{3} + 257 x^{2} - 110 x + 43 \)

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:  \(77541908691589954587750390625=5^{8}\cdot 19^{8}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.83$
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:  $5, 19, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{13} - \frac{2}{5} a^{12} - \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{475} a^{16} - \frac{8}{475} a^{15} - \frac{41}{475} a^{14} - \frac{48}{475} a^{13} - \frac{173}{475} a^{12} - \frac{127}{475} a^{11} + \frac{191}{475} a^{10} - \frac{219}{475} a^{9} - \frac{2}{25} a^{8} - \frac{93}{475} a^{7} - \frac{29}{95} a^{6} + \frac{96}{475} a^{5} - \frac{7}{95} a^{4} - \frac{182}{475} a^{3} + \frac{217}{475} a^{2} + \frac{129}{475} a + \frac{93}{475}$, $\frac{1}{475} a^{17} - \frac{2}{95} a^{15} + \frac{4}{475} a^{14} + \frac{13}{475} a^{13} + \frac{9}{475} a^{12} + \frac{44}{95} a^{11} + \frac{74}{475} a^{10} - \frac{7}{19} a^{9} + \frac{173}{475} a^{8} + \frac{156}{475} a^{7} - \frac{1}{25} a^{6} + \frac{163}{475} a^{5} + \frac{108}{475} a^{4} - \frac{194}{475} a^{3} - \frac{7}{95} a^{2} - \frac{41}{95} a - \frac{16}{475}$, $\frac{1}{28766871625} a^{18} - \frac{9}{28766871625} a^{17} - \frac{3067499}{28766871625} a^{16} + \frac{24540196}{28766871625} a^{15} - \frac{2520231019}{28766871625} a^{14} - \frac{11554708636}{28766871625} a^{13} - \frac{6657835089}{28766871625} a^{12} - \frac{10093297003}{28766871625} a^{11} - \frac{129554803}{1150674865} a^{10} - \frac{1788612426}{28766871625} a^{9} - \frac{5255672569}{28766871625} a^{8} - \frac{542285146}{28766871625} a^{7} - \frac{4678313436}{28766871625} a^{6} - \frac{7863544883}{28766871625} a^{5} + \frac{14201324209}{28766871625} a^{4} - \frac{1671740351}{28766871625} a^{3} + \frac{12679378442}{28766871625} a^{2} + \frac{196063668}{28766871625} a - \frac{12085876988}{28766871625}$, $\frac{1}{4804067561375} a^{19} + \frac{74}{4804067561375} a^{18} + \frac{1450415794}{4804067561375} a^{17} - \frac{2470850116}{4804067561375} a^{16} - \frac{8628409864}{252845661125} a^{15} - \frac{220855006883}{4804067561375} a^{14} + \frac{1162928724703}{4804067561375} a^{13} + \frac{34347952731}{960813512275} a^{12} - \frac{708412664509}{4804067561375} a^{11} + \frac{2078518189164}{4804067561375} a^{10} - \frac{1942525424322}{4804067561375} a^{9} - \frac{1808912603968}{4804067561375} a^{8} + \frac{1438316305396}{4804067561375} a^{7} - \frac{2124053274456}{4804067561375} a^{6} - \frac{156703699181}{960813512275} a^{5} + \frac{635736487766}{4804067561375} a^{4} + \frac{2249341783514}{4804067561375} a^{3} - \frac{1877942159461}{4804067561375} a^{2} + \frac{665643769326}{4804067561375} a + \frac{1987112813121}{4804067561375}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4728089.36014 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:D_5$ (as 20T39):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 10 conjugacy class representatives for $C_2^4:D_5$
Character table for $C_2^4:D_5$

Intermediate fields

5.5.667489.1, 10.6.11138539128025.1, 10.2.278463478200625.1, 10.6.11138539128025.2

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

Sibling fields

Degree 10 siblings: data not computed
Degree 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$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}$
19.2.1.1$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$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$