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Properties

Label 20.8.79692860778...7424.1
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, 10, -38, -572, -1290, 1851, 3205, -3718, 2385, 871, 483, -197, 324, -165, -173, 41, -48, 0, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 2*x^18 - 48*x^16 + 41*x^15 - 173*x^14 - 165*x^13 + 324*x^12 - 197*x^11 + 483*x^10 + 871*x^9 + 2385*x^8 - 3718*x^7 + 3205*x^6 + 1851*x^5 - 1290*x^4 - 572*x^3 - 38*x^2 + 10*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 2*x^18 - 48*x^16 + 41*x^15 - 173*x^14 - 165*x^13 + 324*x^12 - 197*x^11 + 483*x^10 + 871*x^9 + 2385*x^8 - 3718*x^7 + 3205*x^6 + 1851*x^5 - 1290*x^4 - 572*x^3 - 38*x^2 + 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 2 x^{18} - 48 x^{16} + 41 x^{15} - 173 x^{14} - 165 x^{13} + 324 x^{12} - 197 x^{11} + 483 x^{10} + 871 x^{9} + 2385 x^{8} - 3718 x^{7} + 3205 x^{6} + 1851 x^{5} - 1290 x^{4} - 572 x^{3} - 38 x^{2} + 10 x + 1 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2179365836988981028743014873022746509} a^{19} + \frac{305622498871750867423027560358348892}{2179365836988981028743014873022746509} a^{18} + \frac{118553285432632456850173531249740752}{2179365836988981028743014873022746509} a^{17} + \frac{736544633159741350635470135372165113}{2179365836988981028743014873022746509} a^{16} + \frac{11310837955181394656678477118952490}{2179365836988981028743014873022746509} a^{15} - \frac{156787213971909666168284387652560957}{2179365836988981028743014873022746509} a^{14} + \frac{380842740201481557184505728986010177}{2179365836988981028743014873022746509} a^{13} - \frac{974902349660490077657922742344557391}{2179365836988981028743014873022746509} a^{12} - \frac{112067941464578100658171451153195642}{2179365836988981028743014873022746509} a^{11} - \frac{546841444429230994959413393174727173}{2179365836988981028743014873022746509} a^{10} - \frac{549846052388712968552217530557962251}{2179365836988981028743014873022746509} a^{9} - \frac{42625339663266687722239872572542497}{94755036390825262119261516218380283} a^{8} - \frac{90996226496373922013536233929123610}{2179365836988981028743014873022746509} a^{7} - \frac{480512474125840519397044708779063874}{2179365836988981028743014873022746509} a^{6} + \frac{591681525135793848353931665818418105}{2179365836988981028743014873022746509} a^{5} + \frac{274614600549116488339255457084177920}{2179365836988981028743014873022746509} a^{4} - \frac{30674794812355954054969112861839062}{2179365836988981028743014873022746509} a^{3} + \frac{1011495386891385741621969906565380216}{2179365836988981028743014873022746509} a^{2} + \frac{367601062623157437911286845973336872}{2179365836988981028743014873022746509} a - \frac{307589959071596569765076109329686389}{2179365836988981028743014873022746509}$

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:  \( 6746404.9842 \) (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 $20$ $20$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$2.10.10.4$x^{10} - 5 x^{8} + 14 x^{6} - 22 x^{4} + 17 x^{2} - 37$$2$$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
241Data not computed