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Properties

Label 20.8.69654621282...6096.5
Degree $20$
Signature $[8, 6]$
Discriminant $2^{10}\cdot 11^{16}\cdot 23^{6}$
Root discriminant $24.67$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T751

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![89, -1765, 8605, -12537, -5969, 33693, -35499, 20904, -16529, 14480, -6347, 1486, -1515, 1164, -239, 0, -21, -17, 26, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 26*x^18 - 17*x^17 - 21*x^16 - 239*x^14 + 1164*x^13 - 1515*x^12 + 1486*x^11 - 6347*x^10 + 14480*x^9 - 16529*x^8 + 20904*x^7 - 35499*x^6 + 33693*x^5 - 5969*x^4 - 12537*x^3 + 8605*x^2 - 1765*x + 89)
 
gp: K = bnfinit(x^20 - 9*x^19 + 26*x^18 - 17*x^17 - 21*x^16 - 239*x^14 + 1164*x^13 - 1515*x^12 + 1486*x^11 - 6347*x^10 + 14480*x^9 - 16529*x^8 + 20904*x^7 - 35499*x^6 + 33693*x^5 - 5969*x^4 - 12537*x^3 + 8605*x^2 - 1765*x + 89, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 26 x^{18} - 17 x^{17} - 21 x^{16} - 239 x^{14} + 1164 x^{13} - 1515 x^{12} + 1486 x^{11} - 6347 x^{10} + 14480 x^{9} - 16529 x^{8} + 20904 x^{7} - 35499 x^{6} + 33693 x^{5} - 5969 x^{4} - 12537 x^{3} + 8605 x^{2} - 1765 x + 89 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6965462128295683654948996096=2^{10}\cdot 11^{16}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1891198346471965982746146321691937} a^{19} + \frac{856521857076848258539314504208243}{1891198346471965982746146321691937} a^{18} + \frac{661071365162606781887564032354179}{1891198346471965982746146321691937} a^{17} - \frac{822148641274483892941998394459595}{1891198346471965982746146321691937} a^{16} + \frac{284799783422107127390905903096781}{1891198346471965982746146321691937} a^{15} - \frac{872488779590957240712434331281521}{1891198346471965982746146321691937} a^{14} + \frac{858771290867762888519947470739658}{1891198346471965982746146321691937} a^{13} + \frac{900137821420014156851342524623874}{1891198346471965982746146321691937} a^{12} - \frac{597018645282876215881550182669191}{1891198346471965982746146321691937} a^{11} - \frac{424738332641655884827871960291712}{1891198346471965982746146321691937} a^{10} - \frac{695388028882564143834181890477757}{1891198346471965982746146321691937} a^{9} - \frac{197395136541519299357313923986174}{1891198346471965982746146321691937} a^{8} + \frac{433395587285269699132040863301734}{1891198346471965982746146321691937} a^{7} - \frac{945309289577758544895393423333684}{1891198346471965982746146321691937} a^{6} - \frac{183147856951033015872502839132104}{1891198346471965982746146321691937} a^{5} - \frac{833724208990198631727727445474757}{1891198346471965982746146321691937} a^{4} + \frac{707364174302100481990800470146520}{1891198346471965982746146321691937} a^{3} - \frac{268840624284198781117881148188331}{1891198346471965982746146321691937} a^{2} - \frac{243829824317935968055225609399938}{1891198346471965982746146321691937} a - \frac{827492120108977379695505212455995}{1891198346471965982746146321691937}$

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:  $13$
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:  \( 1802003.50084 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T751:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n751 are not computed
Character table for t20n751 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.4930254263.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.10.10.5$x^{10} - 9 x^{8} + 50 x^{6} - 50 x^{4} + 45 x^{2} - 5$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$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}$
$23$$\Q_{23}$$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$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$