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Properties

Label 20.12.1938547717...5625.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{10}\cdot 19^{8}\cdot 43^{8}$
Root discriminant $32.69$
Ramified primes $5, 19, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T73)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-61, 34, 789, -91, -3276, -389, 5511, 1296, -4902, -1347, 2982, 60, -773, 95, -15, 127, -53, -24, 25, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 25*x^18 - 24*x^17 - 53*x^16 + 127*x^15 - 15*x^14 + 95*x^13 - 773*x^12 + 60*x^11 + 2982*x^10 - 1347*x^9 - 4902*x^8 + 1296*x^7 + 5511*x^6 - 389*x^5 - 3276*x^4 - 91*x^3 + 789*x^2 + 34*x - 61)
 
gp: K = bnfinit(x^20 - 8*x^19 + 25*x^18 - 24*x^17 - 53*x^16 + 127*x^15 - 15*x^14 + 95*x^13 - 773*x^12 + 60*x^11 + 2982*x^10 - 1347*x^9 - 4902*x^8 + 1296*x^7 + 5511*x^6 - 389*x^5 - 3276*x^4 - 91*x^3 + 789*x^2 + 34*x - 61, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 25 x^{18} - 24 x^{17} - 53 x^{16} + 127 x^{15} - 15 x^{14} + 95 x^{13} - 773 x^{12} + 60 x^{11} + 2982 x^{10} - 1347 x^{9} - 4902 x^{8} + 1296 x^{7} + 5511 x^{6} - 389 x^{5} - 3276 x^{4} - 91 x^{3} + 789 x^{2} + 34 x - 61 \)

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:  \(1938547717289748864693759765625=5^{10}\cdot 19^{8}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.69$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{13} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{13} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{17} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{95} a^{18} + \frac{8}{95} a^{17} - \frac{1}{19} a^{16} - \frac{4}{95} a^{14} - \frac{32}{95} a^{13} - \frac{9}{95} a^{12} - \frac{28}{95} a^{11} - \frac{8}{95} a^{10} + \frac{24}{95} a^{9} + \frac{32}{95} a^{8} - \frac{2}{19} a^{7} + \frac{28}{95} a^{6} - \frac{39}{95} a^{5} - \frac{12}{95} a^{4} - \frac{24}{95} a^{3} - \frac{16}{95} a^{2} - \frac{13}{95} a - \frac{26}{95}$, $\frac{1}{2861334808890790842802835} a^{19} + \frac{13313269404943934490106}{2861334808890790842802835} a^{18} + \frac{195745930555287987198839}{2861334808890790842802835} a^{17} + \frac{214378776328624135938988}{2861334808890790842802835} a^{16} + \frac{79806247577011864152252}{2861334808890790842802835} a^{15} + \frac{44771388938304156154607}{572266961778158168560567} a^{14} - \frac{786434584037418276250643}{2861334808890790842802835} a^{13} + \frac{776495289659225938595953}{2861334808890790842802835} a^{12} - \frac{1002385371420601164997724}{2861334808890790842802835} a^{11} - \frac{114855395535981443248834}{572266961778158168560567} a^{10} + \frac{692653782058829809483636}{2861334808890790842802835} a^{9} - \frac{194081156644589569676239}{2861334808890790842802835} a^{8} - \frac{1285633629439237106550001}{2861334808890790842802835} a^{7} - \frac{1115857565662304655261459}{2861334808890790842802835} a^{6} - \frac{39908578357027845709812}{572266961778158168560567} a^{5} - \frac{983179439515363786167329}{2861334808890790842802835} a^{4} - \frac{369272901511140072588661}{2861334808890790842802835} a^{3} + \frac{517168144403289309631574}{2861334808890790842802835} a^{2} - \frac{48768834054069596498591}{572266961778158168560567} a + \frac{69932070614906485211600}{572266961778158168560567}$

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:  \( 118577336.062 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T73):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.667489.1, 10.10.1392317391003125.1, 10.6.55692695640125.1, 10.6.11138539128025.1

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 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$5.4.2.1$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}$
5.4.2.1$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}$
$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$$[\ ]_{2}$
$43$43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_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}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$