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Properties

Label 20.0.49338146756...1664.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{16}$
Root discriminant $19.26$
Ramified primes $2, 11$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^4:C_5$ (as 20T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![131, 332, 628, 988, 1057, 872, 676, 340, 140, 110, 56, 82, 81, 20, 20, 2, 1, 6, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 2*x^18 + 6*x^17 + x^16 + 2*x^15 + 20*x^14 + 20*x^13 + 81*x^12 + 82*x^11 + 56*x^10 + 110*x^9 + 140*x^8 + 340*x^7 + 676*x^6 + 872*x^5 + 1057*x^4 + 988*x^3 + 628*x^2 + 332*x + 131)
 
gp: K = bnfinit(x^20 - 2*x^19 + 2*x^18 + 6*x^17 + x^16 + 2*x^15 + 20*x^14 + 20*x^13 + 81*x^12 + 82*x^11 + 56*x^10 + 110*x^9 + 140*x^8 + 340*x^7 + 676*x^6 + 872*x^5 + 1057*x^4 + 988*x^3 + 628*x^2 + 332*x + 131, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 2 x^{18} + 6 x^{17} + x^{16} + 2 x^{15} + 20 x^{14} + 20 x^{13} + 81 x^{12} + 82 x^{11} + 56 x^{10} + 110 x^{9} + 140 x^{8} + 340 x^{7} + 676 x^{6} + 872 x^{5} + 1057 x^{4} + 988 x^{3} + 628 x^{2} + 332 x + 131 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(49338146756019243307761664=2^{30}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.26$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} - \frac{4}{11} a^{13} - \frac{4}{11} a^{12} + \frac{2}{11} a^{11} + \frac{4}{11} a^{10} - \frac{4}{11} a^{9} + \frac{4}{11} a^{8} + \frac{2}{11} a^{7} - \frac{3}{11} a^{6} - \frac{2}{11} a^{5} - \frac{2}{11} a^{4} - \frac{1}{11} a^{3} + \frac{1}{11} a^{2} - \frac{5}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{16} - \frac{4}{11} a^{14} - \frac{4}{11} a^{13} + \frac{2}{11} a^{12} + \frac{4}{11} a^{11} - \frac{4}{11} a^{10} + \frac{4}{11} a^{9} + \frac{2}{11} a^{8} - \frac{3}{11} a^{7} - \frac{2}{11} a^{6} - \frac{2}{11} a^{5} - \frac{1}{11} a^{4} + \frac{1}{11} a^{3} - \frac{5}{11} a^{2} - \frac{1}{11} a$, $\frac{1}{11} a^{17} - \frac{4}{11} a^{14} - \frac{3}{11} a^{13} - \frac{1}{11} a^{12} + \frac{4}{11} a^{11} - \frac{2}{11} a^{10} - \frac{3}{11} a^{9} + \frac{2}{11} a^{8} - \frac{5}{11} a^{7} - \frac{3}{11} a^{6} + \frac{2}{11} a^{5} + \frac{4}{11} a^{4} + \frac{2}{11} a^{3} + \frac{3}{11} a^{2} + \frac{2}{11} a - \frac{4}{11}$, $\frac{1}{253} a^{18} - \frac{1}{253} a^{17} - \frac{5}{253} a^{16} + \frac{7}{253} a^{15} - \frac{100}{253} a^{14} + \frac{4}{23} a^{13} - \frac{49}{253} a^{12} + \frac{29}{253} a^{11} - \frac{102}{253} a^{10} - \frac{125}{253} a^{9} + \frac{60}{253} a^{8} - \frac{5}{11} a^{7} + \frac{125}{253} a^{6} - \frac{87}{253} a^{5} + \frac{3}{11} a^{4} + \frac{106}{253} a^{3} + \frac{13}{253} a^{2} + \frac{43}{253} a - \frac{51}{253}$, $\frac{1}{589444403621731022299} a^{19} - \frac{617391642588283746}{589444403621731022299} a^{18} - \frac{5768284725188774244}{589444403621731022299} a^{17} + \frac{7929187853641615585}{589444403621731022299} a^{16} - \frac{307275416206411597}{25628017548770914013} a^{15} - \frac{28972782711596796672}{589444403621731022299} a^{14} - \frac{20264627127552497031}{589444403621731022299} a^{13} + \frac{162328076493306445387}{589444403621731022299} a^{12} + \frac{7707193356341663207}{53585854874702820209} a^{11} - \frac{294400948071543998453}{589444403621731022299} a^{10} + \frac{136732316605189880658}{589444403621731022299} a^{9} - \frac{169519933002461246399}{589444403621731022299} a^{8} + \frac{194989869568475101212}{589444403621731022299} a^{7} - \frac{9534521318202996556}{25628017548770914013} a^{6} + \frac{11437105477841624712}{53585854874702820209} a^{5} - \frac{1982899899885014366}{53585854874702820209} a^{4} + \frac{231383550097191677087}{589444403621731022299} a^{3} - \frac{168299966909117994433}{589444403621731022299} a^{2} + \frac{25458119932028392103}{53585854874702820209} a + \frac{18129648494721281331}{53585854874702820209}$

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:  $9$
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:  \( 36792.3507976 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T46):

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

\(\Q(\zeta_{11})^+\), 10.4.219503494144.1, 10.4.219503494144.2, 10.2.219503494144.1

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 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$