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Properties

Label 20.4.15709298240...8961.1
Degree $20$
Signature $[4, 8]$
Discriminant $11^{16}\cdot 43^{4}$
Root discriminant $14.45$
Ramified primes $11, 43$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^4:C_5$ (as 20T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -2, 0, 0, -11, -39, -55, -67, -66, -51, -66, -67, -55, -39, -11, 0, 0, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^18 - 11*x^15 - 39*x^14 - 55*x^13 - 67*x^12 - 66*x^11 - 51*x^10 - 66*x^9 - 67*x^8 - 55*x^7 - 39*x^6 - 11*x^5 - 2*x^2 + 1)
 
gp: K = bnfinit(x^20 - 2*x^18 - 11*x^15 - 39*x^14 - 55*x^13 - 67*x^12 - 66*x^11 - 51*x^10 - 66*x^9 - 67*x^8 - 55*x^7 - 39*x^6 - 11*x^5 - 2*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{18} - 11 x^{15} - 39 x^{14} - 55 x^{13} - 67 x^{12} - 66 x^{11} - 51 x^{10} - 66 x^{9} - 67 x^{8} - 55 x^{7} - 39 x^{6} - 11 x^{5} - 2 x^{2} + 1 \)

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:  \(157092982407310367598961=11^{16}\cdot 43^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.45$
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:  $11, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23} a^{17} + \frac{4}{23} a^{16} + \frac{10}{23} a^{15} + \frac{2}{23} a^{14} - \frac{5}{23} a^{13} - \frac{6}{23} a^{12} + \frac{5}{23} a^{11} + \frac{7}{23} a^{10} + \frac{4}{23} a^{9} - \frac{4}{23} a^{8} - \frac{7}{23} a^{7} - \frac{5}{23} a^{6} + \frac{6}{23} a^{5} + \frac{5}{23} a^{4} - \frac{2}{23} a^{3} - \frac{10}{23} a^{2} - \frac{4}{23} a - \frac{1}{23}$, $\frac{1}{7558099} a^{18} + \frac{48437}{7558099} a^{17} - \frac{2125532}{7558099} a^{16} + \frac{996818}{7558099} a^{15} + \frac{1233682}{7558099} a^{14} + \frac{1513350}{7558099} a^{13} + \frac{813062}{7558099} a^{12} + \frac{2417673}{7558099} a^{11} + \frac{3428648}{7558099} a^{10} - \frac{146373}{7558099} a^{9} + \frac{2114196}{7558099} a^{8} + \frac{3732125}{7558099} a^{7} - \frac{1815842}{7558099} a^{6} + \frac{2827802}{7558099} a^{5} + \frac{905069}{7558099} a^{4} + \frac{100490}{328613} a^{3} - \frac{135277}{328613} a^{2} - \frac{1923241}{7558099} a + \frac{2628905}{7558099}$, $\frac{1}{7558099} a^{19} - \frac{3}{7558099} a^{17} - \frac{2677341}{7558099} a^{16} - \frac{831985}{7558099} a^{15} + \frac{2946527}{7558099} a^{14} + \frac{1066570}{7558099} a^{13} - \frac{3485083}{7558099} a^{12} - \frac{155903}{7558099} a^{11} - \frac{446061}{7558099} a^{10} + \frac{843270}{7558099} a^{9} - \frac{2811210}{7558099} a^{8} + \frac{843254}{7558099} a^{7} - \frac{446050}{7558099} a^{6} - \frac{155875}{7558099} a^{5} - \frac{3485039}{7558099} a^{4} + \frac{1066609}{7558099} a^{3} + \frac{2946538}{7558099} a^{2} - \frac{831987}{7558099} a - \frac{2677341}{7558099}$

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:  \( 3897.16327263 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T41):

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

\(\Q(\zeta_{11})^+\), 10.4.9217431883.1 x2, 10.6.396349570969.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$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}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$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}$