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Properties

Label 20.8.62490080598...3125.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{11}\cdot 61^{6}\cdot 397^{4}$
Root discriminant $27.53$
Ramified primes $5, 61, 397$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T802

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![131, -773, 1791, -3268, 6536, -9924, 7813, -470, -3449, 821, 2328, -2816, 2192, -1192, 108, 355, -227, 35, 17, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 17*x^18 + 35*x^17 - 227*x^16 + 355*x^15 + 108*x^14 - 1192*x^13 + 2192*x^12 - 2816*x^11 + 2328*x^10 + 821*x^9 - 3449*x^8 - 470*x^7 + 7813*x^6 - 9924*x^5 + 6536*x^4 - 3268*x^3 + 1791*x^2 - 773*x + 131)
 
gp: K = bnfinit(x^20 - 8*x^19 + 17*x^18 + 35*x^17 - 227*x^16 + 355*x^15 + 108*x^14 - 1192*x^13 + 2192*x^12 - 2816*x^11 + 2328*x^10 + 821*x^9 - 3449*x^8 - 470*x^7 + 7813*x^6 - 9924*x^5 + 6536*x^4 - 3268*x^3 + 1791*x^2 - 773*x + 131, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 17 x^{18} + 35 x^{17} - 227 x^{16} + 355 x^{15} + 108 x^{14} - 1192 x^{13} + 2192 x^{12} - 2816 x^{11} + 2328 x^{10} + 821 x^{9} - 3449 x^{8} - 470 x^{7} + 7813 x^{6} - 9924 x^{5} + 6536 x^{4} - 3268 x^{3} + 1791 x^{2} - 773 x + 131 \)

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:  \(62490080598623484767626953125=5^{11}\cdot 61^{6}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.53$
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:  $5, 61, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{223801475269251989076129532561} a^{19} - \frac{106391393789324233013084256094}{223801475269251989076129532561} a^{18} - \frac{31428122368883687131703713591}{223801475269251989076129532561} a^{17} + \frac{63601229271572534482882232503}{223801475269251989076129532561} a^{16} + \frac{49691096513430354783504359147}{223801475269251989076129532561} a^{15} - \frac{42321508388170663591459192761}{223801475269251989076129532561} a^{14} - \frac{98609410694182047305565343378}{223801475269251989076129532561} a^{13} - \frac{72975439633534154867446586188}{223801475269251989076129532561} a^{12} + \frac{38210336771069890858366494171}{223801475269251989076129532561} a^{11} + \frac{91019296032916810085440049911}{223801475269251989076129532561} a^{10} - \frac{64293934275474340084212305825}{223801475269251989076129532561} a^{9} + \frac{6289836958834358904357244491}{223801475269251989076129532561} a^{8} - \frac{78813541556417371792514916129}{223801475269251989076129532561} a^{7} + \frac{30812181381790704389011375406}{223801475269251989076129532561} a^{6} - \frac{102699068359664400832189974039}{223801475269251989076129532561} a^{5} - \frac{505204516406095807022283976}{7717292250663861692280328709} a^{4} + \frac{3609731251130907633490816060}{223801475269251989076129532561} a^{3} + \frac{38681249946558990959758606017}{223801475269251989076129532561} a^{2} + \frac{16337853122099126479587384149}{223801475269251989076129532561} a - \frac{62337468019535463213228554940}{223801475269251989076129532561}$

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:  \( 7038157.08764 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T802:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n802 are not computed
Character table for t20n802 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.24217.1, 10.10.1832697153125.1

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 $20$ $20$ R $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
61Data not computed
397Data not computed