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Properties

Label 20.0.15558928209...4521.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{18}\cdot 23^{4}$
Root discriminant $16.20$
Ramified primes $11, 23$
Class number $2$
Class group $[2]$
Galois group $C_2\times C_2^4:C_5$ (as 20T44)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 15, -45, 88, -109, 84, -29, -23, 55, -65, 55, -23, -29, 84, -109, 88, -45, 15, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 15*x^18 - 45*x^17 + 88*x^16 - 109*x^15 + 84*x^14 - 29*x^13 - 23*x^12 + 55*x^11 - 65*x^10 + 55*x^9 - 23*x^8 - 29*x^7 + 84*x^6 - 109*x^5 + 88*x^4 - 45*x^3 + 15*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 15*x^18 - 45*x^17 + 88*x^16 - 109*x^15 + 84*x^14 - 29*x^13 - 23*x^12 + 55*x^11 - 65*x^10 + 55*x^9 - 23*x^8 - 29*x^7 + 84*x^6 - 109*x^5 + 88*x^4 - 45*x^3 + 15*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 15 x^{18} - 45 x^{17} + 88 x^{16} - 109 x^{15} + 84 x^{14} - 29 x^{13} - 23 x^{12} + 55 x^{11} - 65 x^{10} + 55 x^{9} - 23 x^{8} - 29 x^{7} + 84 x^{6} - 109 x^{5} + 88 x^{4} - 45 x^{3} + 15 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1555892820924979549874521=11^{18}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.20$
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:  $11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23} a^{17} - \frac{9}{23} a^{16} + \frac{9}{23} a^{15} + \frac{6}{23} a^{13} - \frac{10}{23} a^{12} - \frac{11}{23} a^{11} + \frac{1}{23} a^{10} - \frac{9}{23} a^{9} - \frac{9}{23} a^{8} + \frac{1}{23} a^{7} - \frac{11}{23} a^{6} - \frac{10}{23} a^{5} + \frac{6}{23} a^{4} + \frac{9}{23} a^{2} - \frac{9}{23} a + \frac{1}{23}$, $\frac{1}{23} a^{18} - \frac{3}{23} a^{16} - \frac{11}{23} a^{15} + \frac{6}{23} a^{14} - \frac{2}{23} a^{13} - \frac{9}{23} a^{12} - \frac{6}{23} a^{11} + \frac{2}{23} a^{9} - \frac{11}{23} a^{8} - \frac{2}{23} a^{7} + \frac{6}{23} a^{6} + \frac{8}{23} a^{5} + \frac{8}{23} a^{4} + \frac{9}{23} a^{3} + \frac{3}{23} a^{2} - \frac{11}{23} a + \frac{9}{23}$, $\frac{1}{23} a^{19} + \frac{8}{23} a^{16} + \frac{10}{23} a^{15} - \frac{2}{23} a^{14} + \frac{9}{23} a^{13} + \frac{10}{23} a^{12} - \frac{10}{23} a^{11} + \frac{5}{23} a^{10} + \frac{8}{23} a^{9} - \frac{6}{23} a^{8} + \frac{9}{23} a^{7} - \frac{2}{23} a^{6} + \frac{1}{23} a^{5} + \frac{4}{23} a^{4} + \frac{3}{23} a^{3} - \frac{7}{23} a^{2} + \frac{5}{23} a + \frac{3}{23}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{924}{23} a^{19} + \frac{3305}{23} a^{18} - \frac{12048}{23} a^{17} + \frac{35580}{23} a^{16} - \frac{62119}{23} a^{15} + \frac{64274}{23} a^{14} - \frac{36752}{23} a^{13} + \frac{1455}{23} a^{12} + \frac{24786}{23} a^{11} - \frac{36977}{23} a^{10} + \frac{37983}{23} a^{9} - \frac{27098}{23} a^{8} + \frac{3386}{23} a^{7} + \frac{31501}{23} a^{6} - \frac{61280}{23} a^{5} + \frac{64904}{23} a^{4} - \frac{40164}{23} a^{3} + \frac{14533}{23} a^{2} - \frac{3659}{23} a + \frac{1263}{23} \) (order $22$)
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:  \( 38489.7797463 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T44):

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

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{11})^+\), 10.8.1247354328539.1, \(\Q(\zeta_{11})\), 10.2.113395848049.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$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$$0$Trivial$[\ ]$
$\Q_{23}$$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}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$