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Properties

Label 20.10.1807948901...9712.3
Degree $20$
Signature $[10, 5]$
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, 14, 62, 57, -267, -639, -74, 803, 94, -622, 115, -19, -279, 385, 170, -276, -26, 72, -5, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 5*x^18 + 72*x^17 - 26*x^16 - 276*x^15 + 170*x^14 + 385*x^13 - 279*x^12 - 19*x^11 + 115*x^10 - 622*x^9 + 94*x^8 + 803*x^7 - 74*x^6 - 639*x^5 - 267*x^4 + 57*x^3 + 62*x^2 + 14*x + 1)
 
gp: K = bnfinit(x^20 - 6*x^19 - 5*x^18 + 72*x^17 - 26*x^16 - 276*x^15 + 170*x^14 + 385*x^13 - 279*x^12 - 19*x^11 + 115*x^10 - 622*x^9 + 94*x^8 + 803*x^7 - 74*x^6 - 639*x^5 - 267*x^4 + 57*x^3 + 62*x^2 + 14*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 5 x^{18} + 72 x^{17} - 26 x^{16} - 276 x^{15} + 170 x^{14} + 385 x^{13} - 279 x^{12} - 19 x^{11} + 115 x^{10} - 622 x^{9} + 94 x^{8} + 803 x^{7} - 74 x^{6} - 639 x^{5} - 267 x^{4} + 57 x^{3} + 62 x^{2} + 14 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{195316874103837454615613} a^{19} - \frac{71210044416181937480932}{195316874103837454615613} a^{18} - \frac{64606546298780045594187}{195316874103837454615613} a^{17} - \frac{9957327943513112241957}{195316874103837454615613} a^{16} + \frac{66882826319828061278275}{195316874103837454615613} a^{15} - \frac{5339903531735835301066}{195316874103837454615613} a^{14} - \frac{35353335707522733379192}{195316874103837454615613} a^{13} - \frac{95232958904331483219508}{195316874103837454615613} a^{12} - \frac{9113668633508539204771}{195316874103837454615613} a^{11} - \frac{37594722494982550382819}{195316874103837454615613} a^{10} - \frac{90610380837775212646404}{195316874103837454615613} a^{9} - \frac{73917756345482803994288}{195316874103837454615613} a^{8} - \frac{81886817675833681640953}{195316874103837454615613} a^{7} - \frac{87065068599323043727073}{195316874103837454615613} a^{6} - \frac{95103167445838258512494}{195316874103837454615613} a^{5} + \frac{28928836645501560941237}{195316874103837454615613} a^{4} - \frac{90796888598501281730747}{195316874103837454615613} a^{3} - \frac{25555450848027575533796}{195316874103837454615613} a^{2} + \frac{88259747962615322787929}{195316874103837454615613} a - \frac{94500884364261373792}{195316874103837454615613}$

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:  $14$
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:  \( 4241933.16948 \) (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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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