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Properties

Label 20.6.26101424948...8128.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{40}\cdot 3^{16}\cdot 223^{5}$
Root discriminant $37.22$
Ramified primes $2, 3, 223$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 24, -72, -412, 956, 584, -1360, -831, 1024, 1158, -336, -984, -352, 178, 176, 36, -16, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 - 16*x^17 + 36*x^16 + 176*x^15 + 178*x^14 - 352*x^13 - 984*x^12 - 336*x^11 + 1158*x^10 + 1024*x^9 - 831*x^8 - 1360*x^7 + 584*x^6 + 956*x^5 - 412*x^4 - 72*x^3 + 24*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^18 - 16*x^17 + 36*x^16 + 176*x^15 + 178*x^14 - 352*x^13 - 984*x^12 - 336*x^11 + 1158*x^10 + 1024*x^9 - 831*x^8 - 1360*x^7 + 584*x^6 + 956*x^5 - 412*x^4 - 72*x^3 + 24*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{18} - 16 x^{17} + 36 x^{16} + 176 x^{15} + 178 x^{14} - 352 x^{13} - 984 x^{12} - 336 x^{11} + 1158 x^{10} + 1024 x^{9} - 831 x^{8} - 1360 x^{7} + 584 x^{6} + 956 x^{5} - 412 x^{4} - 72 x^{3} + 24 x^{2} - 4 x + 1 \)

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:  \(-26101424948512167208849797808128=-\,2^{40}\cdot 3^{16}\cdot 223^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.22$
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:  $2, 3, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{641} a^{18} - \frac{67}{641} a^{17} + \frac{135}{641} a^{16} - \frac{53}{641} a^{15} - \frac{184}{641} a^{14} - \frac{203}{641} a^{13} + \frac{287}{641} a^{12} + \frac{106}{641} a^{11} + \frac{264}{641} a^{10} - \frac{302}{641} a^{9} + \frac{172}{641} a^{8} + \frac{158}{641} a^{7} - \frac{282}{641} a^{6} - \frac{255}{641} a^{5} - \frac{222}{641} a^{4} - \frac{123}{641} a^{3} - \frac{200}{641} a^{2} + \frac{226}{641} a - \frac{130}{641}$, $\frac{1}{1418885408341506102302539} a^{19} + \frac{443578091003076320852}{1418885408341506102302539} a^{18} + \frac{169775332097864039057947}{1418885408341506102302539} a^{17} + \frac{196398613433567322508872}{1418885408341506102302539} a^{16} - \frac{148724656972806582394318}{1418885408341506102302539} a^{15} + \frac{165070190018149641778013}{1418885408341506102302539} a^{14} - \frac{1195077870274365195233}{8201649759199457238743} a^{13} - \frac{385438534661238173944125}{1418885408341506102302539} a^{12} + \frac{517204042656852156862173}{1418885408341506102302539} a^{11} - \frac{598855357421988829987040}{1418885408341506102302539} a^{10} - \frac{99913452127665626189729}{1418885408341506102302539} a^{9} + \frac{661305797171628241049085}{1418885408341506102302539} a^{8} + \frac{33976248467906973844639}{109145031410885084792503} a^{7} + \frac{260435777271629523638534}{1418885408341506102302539} a^{6} + \frac{493423129712393888162372}{1418885408341506102302539} a^{5} + \frac{563982225077803700608140}{1418885408341506102302539} a^{4} + \frac{269147209372944866709007}{1418885408341506102302539} a^{3} + \frac{206636801770728363699740}{1418885408341506102302539} a^{2} + \frac{627613499666568876700907}{1418885408341506102302539} a - \frac{245308801923906640780309}{1418885408341506102302539}$

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:  \( 89058324.741 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n797 are not computed
Character table for t20n797 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.3.18063.1, 10.6.10691279880192.1

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
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 R ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
2Data not computed
3Data not computed
223Data not computed