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Properties

Label 20.6.18079489015...9712.7
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{10}\cdot 11^{16}\cdot 727^{3}$
Root discriminant $25.87$
Ramified primes $2, 11, 727$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T747

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 16, -101, 280, 49, -981, 2302, -2217, 1189, -639, 357, -697, 1040, -1076, 937, -603, 307, -125, 37, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 37*x^18 - 125*x^17 + 307*x^16 - 603*x^15 + 937*x^14 - 1076*x^13 + 1040*x^12 - 697*x^11 + 357*x^10 - 639*x^9 + 1189*x^8 - 2217*x^7 + 2302*x^6 - 981*x^5 + 49*x^4 + 280*x^3 - 101*x^2 + 16*x - 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 37*x^18 - 125*x^17 + 307*x^16 - 603*x^15 + 937*x^14 - 1076*x^13 + 1040*x^12 - 697*x^11 + 357*x^10 - 639*x^9 + 1189*x^8 - 2217*x^7 + 2302*x^6 - 981*x^5 + 49*x^4 + 280*x^3 - 101*x^2 + 16*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 37 x^{18} - 125 x^{17} + 307 x^{16} - 603 x^{15} + 937 x^{14} - 1076 x^{13} + 1040 x^{12} - 697 x^{11} + 357 x^{10} - 639 x^{9} + 1189 x^{8} - 2217 x^{7} + 2302 x^{6} - 981 x^{5} + 49 x^{4} + 280 x^{3} - 101 x^{2} + 16 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-18079489015266793067734899712=-\,2^{10}\cdot 11^{16}\cdot 727^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.87$
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:  $2, 11, 727$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} - \frac{10}{23} a^{17} + \frac{10}{23} a^{16} + \frac{3}{23} a^{15} - \frac{8}{23} a^{14} + \frac{8}{23} a^{13} + \frac{9}{23} a^{12} + \frac{2}{23} a^{11} - \frac{8}{23} a^{10} + \frac{7}{23} a^{9} + \frac{6}{23} a^{8} + \frac{9}{23} a^{7} - \frac{8}{23} a^{6} - \frac{2}{23} a^{5} - \frac{9}{23} a^{4} + \frac{5}{23} a^{3} + \frac{2}{23} a^{2} - \frac{5}{23} a - \frac{1}{23}$, $\frac{1}{1511096338095175496887662991} a^{19} + \frac{22560711705438088897918}{1511096338095175496887662991} a^{18} - \frac{22613445369737631311568983}{65699840786746760734246217} a^{17} - \frac{423823174150542103381622100}{1511096338095175496887662991} a^{16} - \frac{127948228840724108823186251}{1511096338095175496887662991} a^{15} - \frac{17136512951880781466177374}{65699840786746760734246217} a^{14} + \frac{171328256049617657360383977}{1511096338095175496887662991} a^{13} + \frac{440525690679563594781591343}{1511096338095175496887662991} a^{12} + \frac{686619275594731430332294246}{1511096338095175496887662991} a^{11} - \frac{551827450896057727714569484}{1511096338095175496887662991} a^{10} + \frac{524504067032866749697812708}{1511096338095175496887662991} a^{9} + \frac{94259445230987389930420906}{1511096338095175496887662991} a^{8} - \frac{608722638820928514673132981}{1511096338095175496887662991} a^{7} + \frac{686132037236608990257008497}{1511096338095175496887662991} a^{6} - \frac{510823270569722763106832555}{1511096338095175496887662991} a^{5} + \frac{608811696463404870928194179}{1511096338095175496887662991} a^{4} - \frac{63906837542256118335089723}{1511096338095175496887662991} a^{3} - \frac{347354166380301695630653253}{1511096338095175496887662991} a^{2} - \frac{10308605508846395026185942}{1511096338095175496887662991} a - \frac{349902957722194590233869359}{1511096338095175496887662991}$

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:  $12$
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:  \( 1961288.37288 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T747:

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 t20n747 are not computed
Character table for t20n747 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.155838906487.1

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 $20$ $20$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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$2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$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}$
11Data not computed
727Data not computed