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Properties

Label 20.4.59699157574...1344.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{30}\cdot 11^{18}$
Root discriminant $24.48$
Ramified primes $2, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:C_5$ (as 20T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 30, -56, 33, 144, -536, 1152, -1970, 2992, -3596, 2992, -1970, 1152, -536, 144, 33, -56, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 30*x^18 - 56*x^17 + 33*x^16 + 144*x^15 - 536*x^14 + 1152*x^13 - 1970*x^12 + 2992*x^11 - 3596*x^10 + 2992*x^9 - 1970*x^8 + 1152*x^7 - 536*x^6 + 144*x^5 + 33*x^4 - 56*x^3 + 30*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 30*x^18 - 56*x^17 + 33*x^16 + 144*x^15 - 536*x^14 + 1152*x^13 - 1970*x^12 + 2992*x^11 - 3596*x^10 + 2992*x^9 - 1970*x^8 + 1152*x^7 - 536*x^6 + 144*x^5 + 33*x^4 - 56*x^3 + 30*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 30 x^{18} - 56 x^{17} + 33 x^{16} + 144 x^{15} - 536 x^{14} + 1152 x^{13} - 1970 x^{12} + 2992 x^{11} - 3596 x^{10} + 2992 x^{9} - 1970 x^{8} + 1152 x^{7} - 536 x^{6} + 144 x^{5} + 33 x^{4} - 56 x^{3} + 30 x^{2} - 8 x + 1 \)

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:  \(5969915757478328440239161344=2^{30}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.48$
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:  $2, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} + \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{10} - \frac{1}{8} a^{6} + \frac{1}{8} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{11} - \frac{1}{8} a^{7} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{8} + \frac{1}{16}$, $\frac{1}{16} a^{17} - \frac{1}{8} a^{9} + \frac{1}{16} a$, $\frac{1}{5348656} a^{18} - \frac{10817}{2674328} a^{17} - \frac{150487}{5348656} a^{16} - \frac{25467}{668582} a^{15} + \frac{47469}{1337164} a^{14} - \frac{146217}{2674328} a^{13} - \frac{64933}{2674328} a^{12} + \frac{3920}{334291} a^{11} + \frac{149027}{2674328} a^{10} + \frac{11765}{2674328} a^{9} - \frac{23158}{334291} a^{8} + \frac{3920}{334291} a^{7} + \frac{134679}{1337164} a^{6} + \frac{522365}{2674328} a^{5} + \frac{429229}{2674328} a^{4} - \frac{25467}{668582} a^{3} - \frac{150487}{5348656} a^{2} + \frac{124007}{334291} a + \frac{668583}{5348656}$, $\frac{1}{123019088} a^{19} - \frac{9}{123019088} a^{18} - \frac{490605}{61509544} a^{17} + \frac{840647}{123019088} a^{16} + \frac{378573}{15377386} a^{15} + \frac{418739}{7688693} a^{14} + \frac{118493}{2674328} a^{13} + \frac{774445}{30754772} a^{12} - \frac{2307449}{61509544} a^{11} + \frac{489691}{61509544} a^{10} - \frac{1834045}{61509544} a^{9} - \frac{1830751}{61509544} a^{8} - \frac{1861272}{7688693} a^{7} - \frac{12532}{334291} a^{6} - \frac{6140647}{61509544} a^{5} - \frac{2481507}{30754772} a^{4} - \frac{54809831}{123019088} a^{3} + \frac{33783267}{123019088} a^{2} + \frac{5976703}{30754772} a + \frac{22084831}{123019088}$

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:  \( 894928.598953 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T40):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 16 conjugacy class representatives for $C_2\times C_2^4:C_5$
Character table for $C_2\times C_2^4:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.2414538435584.1, 10.2.2414538435584.1, 10.6.219503494144.2

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 20 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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}$