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Properties

Label 20.20.1670627195...6304.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 11^{18}\cdot 23^{4}$
Root discriminant $45.83$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2\times C_2^4:C_5$ (as 20T74)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![58147, 38132, -422874, -221306, 956087, 485792, -1030838, -519716, 621926, 305624, -227556, -104808, 52829, 21338, -7938, -2520, 759, 158, -42, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 42*x^18 + 158*x^17 + 759*x^16 - 2520*x^15 - 7938*x^14 + 21338*x^13 + 52829*x^12 - 104808*x^11 - 227556*x^10 + 305624*x^9 + 621926*x^8 - 519716*x^7 - 1030838*x^6 + 485792*x^5 + 956087*x^4 - 221306*x^3 - 422874*x^2 + 38132*x + 58147)
 
gp: K = bnfinit(x^20 - 4*x^19 - 42*x^18 + 158*x^17 + 759*x^16 - 2520*x^15 - 7938*x^14 + 21338*x^13 + 52829*x^12 - 104808*x^11 - 227556*x^10 + 305624*x^9 + 621926*x^8 - 519716*x^7 - 1030838*x^6 + 485792*x^5 + 956087*x^4 - 221306*x^3 - 422874*x^2 + 38132*x + 58147, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 42 x^{18} + 158 x^{17} + 759 x^{16} - 2520 x^{15} - 7938 x^{14} + 21338 x^{13} + 52829 x^{12} - 104808 x^{11} - 227556 x^{10} + 305624 x^{9} + 621926 x^{8} - 519716 x^{7} - 1030838 x^{6} + 485792 x^{5} + 956087 x^{4} - 221306 x^{3} - 422874 x^{2} + 38132 x + 58147 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1670627195488492909044967149666304=2^{30}\cdot 11^{18}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.83$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} + \frac{3}{23} a^{17} - \frac{10}{23} a^{16} + \frac{6}{23} a^{15} + \frac{1}{23} a^{14} - \frac{9}{23} a^{13} - \frac{9}{23} a^{12} - \frac{7}{23} a^{11} + \frac{11}{23} a^{10} + \frac{3}{23} a^{9} + \frac{10}{23} a^{8} + \frac{11}{23} a^{7} + \frac{9}{23} a^{6} - \frac{8}{23} a^{5} - \frac{4}{23} a^{4} + \frac{8}{23} a^{3} - \frac{11}{23} a^{2} + \frac{11}{23} a + \frac{6}{23}$, $\frac{1}{621361644313601237747593824161} a^{19} - \frac{4754509562747080589709343398}{621361644313601237747593824161} a^{18} - \frac{1322866717738326829270791706}{5700565544161479245390769029} a^{17} + \frac{86939586066571457105529132935}{621361644313601237747593824161} a^{16} - \frac{201247446014075683717099024904}{621361644313601237747593824161} a^{15} + \frac{8939095642583481571150987055}{27015723665808749467286688007} a^{14} + \frac{82244663783305017636351534193}{621361644313601237747593824161} a^{13} + \frac{129684416368534501265726821698}{621361644313601237747593824161} a^{12} + \frac{283714223174942138498794172493}{621361644313601237747593824161} a^{11} - \frac{98365099213883040598837895212}{621361644313601237747593824161} a^{10} - \frac{267178558712795456818412718477}{621361644313601237747593824161} a^{9} - \frac{90146863688331632980043326072}{621361644313601237747593824161} a^{8} - \frac{269190557193898627986868312674}{621361644313601237747593824161} a^{7} - \frac{23078232929086508869694298242}{621361644313601237747593824161} a^{6} + \frac{290990409091385657462036724811}{621361644313601237747593824161} a^{5} + \frac{294436592429231828350389764915}{621361644313601237747593824161} a^{4} - \frac{75738605688564830972720062504}{621361644313601237747593824161} a^{3} - \frac{305693966199826193944146689157}{621361644313601237747593824161} a^{2} - \frac{28102688417308358563925772089}{621361644313601237747593824161} a - \frac{159147874011152434586068759879}{621361644313601237747593824161}$

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:  $19$
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:  \( 16195762579.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T74):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.1277290832423936.2, 10.10.55534384018432.1, 10.10.5048580365312.1

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
Arithmetically equvalently 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$