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Properties

Label 20.16.5844100138...0000.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{16}\cdot 5^{17}\cdot 43^{8}$
Root discriminant $30.78$
Ramified primes $2, 5, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T135

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -27, 225, -599, -748, 5923, -7759, -2983, 14130, -8977, -3649, 7715, -3092, -1093, 1301, -295, -118, 81, -7, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 7*x^18 + 81*x^17 - 118*x^16 - 295*x^15 + 1301*x^14 - 1093*x^13 - 3092*x^12 + 7715*x^11 - 3649*x^10 - 8977*x^9 + 14130*x^8 - 2983*x^7 - 7759*x^6 + 5923*x^5 - 748*x^4 - 599*x^3 + 225*x^2 - 27*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 - 7*x^18 + 81*x^17 - 118*x^16 - 295*x^15 + 1301*x^14 - 1093*x^13 - 3092*x^12 + 7715*x^11 - 3649*x^10 - 8977*x^9 + 14130*x^8 - 2983*x^7 - 7759*x^6 + 5923*x^5 - 748*x^4 - 599*x^3 + 225*x^2 - 27*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 7 x^{18} + 81 x^{17} - 118 x^{16} - 295 x^{15} + 1301 x^{14} - 1093 x^{13} - 3092 x^{12} + 7715 x^{11} - 3649 x^{10} - 8977 x^{9} + 14130 x^{8} - 2983 x^{7} - 7759 x^{6} + 5923 x^{5} - 748 x^{4} - 599 x^{3} + 225 x^{2} - 27 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(584410013880050000000000000000=2^{16}\cdot 5^{17}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.78$
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:  $2, 5, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} + \frac{2}{5} a^{17} + \frac{1}{5} a^{16} + \frac{1}{5} a^{15} - \frac{2}{5} a^{14} - \frac{2}{5} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{15236357201138982150958255} a^{19} - \frac{189653358515359716474201}{15236357201138982150958255} a^{18} + \frac{1190561450699026734735193}{3047271440227796430191651} a^{17} + \frac{2797485155142213868814878}{15236357201138982150958255} a^{16} - \frac{141843548272702043426030}{3047271440227796430191651} a^{15} + \frac{2991952054197320972407041}{15236357201138982150958255} a^{14} - \frac{5419635131807796720368207}{15236357201138982150958255} a^{13} - \frac{1711816536177422407171}{48991502254466180549705} a^{12} - \frac{3896277942848290363254298}{15236357201138982150958255} a^{11} + \frac{7095278491906116816566046}{15236357201138982150958255} a^{10} + \frac{2509135637909802089302051}{15236357201138982150958255} a^{9} + \frac{4260114919363993943506166}{15236357201138982150958255} a^{8} + \frac{1400066975026404644685002}{15236357201138982150958255} a^{7} - \frac{1418457212946920687920822}{3047271440227796430191651} a^{6} + \frac{5916213267144532613450078}{15236357201138982150958255} a^{5} - \frac{6214247948777829935974674}{15236357201138982150958255} a^{4} + \frac{1032665626281482683506812}{15236357201138982150958255} a^{3} + \frac{3119852814140704639296038}{15236357201138982150958255} a^{2} + \frac{2300177139180854843296686}{15236357201138982150958255} a + \frac{4761253999556515156363657}{15236357201138982150958255}$

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:  $17$
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:  \( 164050260.109 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T135:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 22 conjugacy class representatives for t20n135
Character table for t20n135 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.3698000.1, 10.10.68376020000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$43$43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$