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Properties

Label 20.8.79692860778...7424.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{10}\cdot 11^{18}\cdot 241^{3}$
Root discriminant $27.87$
Ramified primes $2, 11, 241$
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, 13, -196, 198, -950, -580, 1257, -968, -666, 1693, 505, -1443, 559, 396, -722, 371, -25, -66, 39, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 39*x^18 - 66*x^17 - 25*x^16 + 371*x^15 - 722*x^14 + 396*x^13 + 559*x^12 - 1443*x^11 + 505*x^10 + 1693*x^9 - 666*x^8 - 968*x^7 + 1257*x^6 - 580*x^5 - 950*x^4 + 198*x^3 - 196*x^2 + 13*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 39*x^18 - 66*x^17 - 25*x^16 + 371*x^15 - 722*x^14 + 396*x^13 + 559*x^12 - 1443*x^11 + 505*x^10 + 1693*x^9 - 666*x^8 - 968*x^7 + 1257*x^6 - 580*x^5 - 950*x^4 + 198*x^3 - 196*x^2 + 13*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 39 x^{18} - 66 x^{17} - 25 x^{16} + 371 x^{15} - 722 x^{14} + 396 x^{13} + 559 x^{12} - 1443 x^{11} + 505 x^{10} + 1693 x^{9} - 666 x^{8} - 968 x^{7} + 1257 x^{6} - 580 x^{5} - 950 x^{4} + 198 x^{3} - 196 x^{2} + 13 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(79692860778363999735970407424=2^{10}\cdot 11^{18}\cdot 241^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.87$
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:  $2, 11, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{788621001867743121083023227298882} a^{19} - \frac{63730020232073353867046295252949}{394310500933871560541511613649441} a^{18} - \frac{67329740511025904812576356865187}{788621001867743121083023227298882} a^{17} + \frac{4523829647615445582742613948586}{394310500933871560541511613649441} a^{16} + \frac{6290263084625705155601307163427}{394310500933871560541511613649441} a^{15} + \frac{824494876075440143094444932208}{394310500933871560541511613649441} a^{14} - \frac{272151503137319845504005497077133}{788621001867743121083023227298882} a^{13} + \frac{23585417222667002748824925371551}{394310500933871560541511613649441} a^{12} - \frac{261564275380346733502815780973583}{788621001867743121083023227298882} a^{11} + \frac{169468675874373540584667090646529}{788621001867743121083023227298882} a^{10} + \frac{168024962325326012055558966284495}{788621001867743121083023227298882} a^{9} - \frac{151421185392736125522576127741848}{394310500933871560541511613649441} a^{8} - \frac{163413801068076410040244308480531}{788621001867743121083023227298882} a^{7} + \frac{24567128891519563100577158007167}{394310500933871560541511613649441} a^{6} + \frac{173061293796174508194931120686038}{394310500933871560541511613649441} a^{5} - \frac{215113273129711992315789287209317}{788621001867743121083023227298882} a^{4} + \frac{89744387238982421834258863151399}{394310500933871560541511613649441} a^{3} + \frac{174836954574005731192400778927548}{394310500933871560541511613649441} a^{2} + \frac{64271052859111830756137208954706}{394310500933871560541511613649441} a - \frac{107896755286301194674070712689831}{788621001867743121083023227298882}$

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:  \( 5776074.9541 \) (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

\(\Q(\zeta_{11})^+\), 10.6.51660490321.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R $20$ $20$ $20$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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}$
2.10.10.3$x^{10} - 9 x^{8} + 22 x^{6} - 46 x^{4} + 9 x^{2} - 9$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
11Data not computed
241Data not computed