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Properties

Label 20.0.32473988377...0464.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 11^{17}\cdot 89^{4}$
Root discriminant $26.64$
Ramified primes $2, 11, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T427

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![989, -2421, 2687, -3077, 4805, -4727, 3842, -4711, 4289, -3047, 2762, -2131, 1101, -503, 279, -119, 38, -26, 19, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 19*x^18 - 26*x^17 + 38*x^16 - 119*x^15 + 279*x^14 - 503*x^13 + 1101*x^12 - 2131*x^11 + 2762*x^10 - 3047*x^9 + 4289*x^8 - 4711*x^7 + 3842*x^6 - 4727*x^5 + 4805*x^4 - 3077*x^3 + 2687*x^2 - 2421*x + 989)
 
gp: K = bnfinit(x^20 - 7*x^19 + 19*x^18 - 26*x^17 + 38*x^16 - 119*x^15 + 279*x^14 - 503*x^13 + 1101*x^12 - 2131*x^11 + 2762*x^10 - 3047*x^9 + 4289*x^8 - 4711*x^7 + 3842*x^6 - 4727*x^5 + 4805*x^4 - 3077*x^3 + 2687*x^2 - 2421*x + 989, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 19 x^{18} - 26 x^{17} + 38 x^{16} - 119 x^{15} + 279 x^{14} - 503 x^{13} + 1101 x^{12} - 2131 x^{11} + 2762 x^{10} - 3047 x^{9} + 4289 x^{8} - 4711 x^{7} + 3842 x^{6} - 4727 x^{5} + 4805 x^{4} - 3077 x^{3} + 2687 x^{2} - 2421 x + 989 \)

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:  \(32473988377432635504517950464=2^{10}\cdot 11^{17}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.64$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{43} a^{18} + \frac{8}{43} a^{17} - \frac{13}{43} a^{16} - \frac{18}{43} a^{15} - \frac{19}{43} a^{14} + \frac{10}{43} a^{13} + \frac{6}{43} a^{12} + \frac{2}{43} a^{11} + \frac{4}{43} a^{10} - \frac{10}{43} a^{9} - \frac{17}{43} a^{8} - \frac{19}{43} a^{7} + \frac{9}{43} a^{6} - \frac{11}{43} a^{5} - \frac{13}{43} a^{4} + \frac{18}{43} a^{3} - \frac{1}{43} a^{2} + \frac{20}{43} a$, $\frac{1}{558905550048169277634846281864417} a^{19} + \frac{3002354909266583721541919655098}{558905550048169277634846281864417} a^{18} + \frac{50049199277614309551627884415085}{558905550048169277634846281864417} a^{17} - \frac{50117802240019310878942495466555}{558905550048169277634846281864417} a^{16} + \frac{239910015722359398259765699270264}{558905550048169277634846281864417} a^{15} - \frac{125742508490126882283043729849382}{558905550048169277634846281864417} a^{14} + \frac{59496479046323325740824591511930}{558905550048169277634846281864417} a^{13} + \frac{94511063597186328325361852188626}{558905550048169277634846281864417} a^{12} + \frac{220653776249299895610656417738432}{558905550048169277634846281864417} a^{11} + \frac{259714965909561401062644725089860}{558905550048169277634846281864417} a^{10} + \frac{254144727200497349530487263258101}{558905550048169277634846281864417} a^{9} - \frac{250288595548454188867142320335079}{558905550048169277634846281864417} a^{8} - \frac{37532153876779949098158835236737}{558905550048169277634846281864417} a^{7} - \frac{236113160524032998642679131709665}{558905550048169277634846281864417} a^{6} - \frac{163733396456208234518905132773698}{558905550048169277634846281864417} a^{5} + \frac{217215319698065787746641478187087}{558905550048169277634846281864417} a^{4} - \frac{140708911639325833145030223689395}{558905550048169277634846281864417} a^{3} + \frac{1360869565301371018257312355555}{558905550048169277634846281864417} a^{2} - \frac{129686864320035356654342727336795}{558905550048169277634846281864417} a - \frac{3938203756913417662766166557022}{12997803489492308782205727485219}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 472233.92586 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T427:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n427 are not computed
Character table for t20n427 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.19077940409.1

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
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 $20$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
89Data not computed