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Properties

Label 20.0.11230152347...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 60662149^{2}$
Root discriminant $20.07$
Ramified primes $5, 60662149$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T654

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -40, 160, -172, 297, -322, 514, -558, 752, -806, 908, -848, 722, -532, 349, -199, 99, -41, 15, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 15*x^18 - 41*x^17 + 99*x^16 - 199*x^15 + 349*x^14 - 532*x^13 + 722*x^12 - 848*x^11 + 908*x^10 - 806*x^9 + 752*x^8 - 558*x^7 + 514*x^6 - 322*x^5 + 297*x^4 - 172*x^3 + 160*x^2 - 40*x + 16)
 
gp: K = bnfinit(x^20 - 4*x^19 + 15*x^18 - 41*x^17 + 99*x^16 - 199*x^15 + 349*x^14 - 532*x^13 + 722*x^12 - 848*x^11 + 908*x^10 - 806*x^9 + 752*x^8 - 558*x^7 + 514*x^6 - 322*x^5 + 297*x^4 - 172*x^3 + 160*x^2 - 40*x + 16, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 15 x^{18} - 41 x^{17} + 99 x^{16} - 199 x^{15} + 349 x^{14} - 532 x^{13} + 722 x^{12} - 848 x^{11} + 908 x^{10} - 806 x^{9} + 752 x^{8} - 558 x^{7} + 514 x^{6} - 322 x^{5} + 297 x^{4} - 172 x^{3} + 160 x^{2} - 40 x + 16 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(112301523477117950439453125=5^{15}\cdot 60662149^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.07$
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:  $5, 60662149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} + \frac{1}{4} a^{13} + \frac{1}{4} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{927845029981902181528} a^{19} + \frac{1358904378208313557}{115980628747737772691} a^{18} - \frac{228678946884137306693}{927845029981902181528} a^{17} - \frac{350670445972130391685}{927845029981902181528} a^{16} - \frac{201169921862369368725}{927845029981902181528} a^{15} + \frac{279083329924400214977}{927845029981902181528} a^{14} + \frac{77511082571503953037}{927845029981902181528} a^{13} + \frac{65830123838274309323}{231961257495475545382} a^{12} + \frac{44807937755023945843}{463922514990951090764} a^{11} - \frac{14081012088322120319}{115980628747737772691} a^{10} + \frac{115260672655328913917}{231961257495475545382} a^{9} + \frac{207303280174819813577}{463922514990951090764} a^{8} + \frac{17782011700928234419}{115980628747737772691} a^{7} + \frac{174687293668096746005}{463922514990951090764} a^{6} + \frac{3624829636360458001}{463922514990951090764} a^{5} + \frac{121174719607637151451}{463922514990951090764} a^{4} + \frac{373334533658927887561}{927845029981902181528} a^{3} + \frac{29601742789259551419}{115980628747737772691} a^{2} + \frac{92605649223106940141}{231961257495475545382} a - \frac{55265434337172998926}{115980628747737772691}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{23758269720454200313}{927845029981902181528} a^{19} - \frac{19727170792655448601}{231961257495475545382} a^{18} + \frac{298908664819677718983}{927845029981902181528} a^{17} - \frac{759331132788090166505}{927845029981902181528} a^{16} + \frac{1786371607278417965267}{927845029981902181528} a^{15} - \frac{3396230831720009486687}{927845029981902181528} a^{14} + \frac{5697308346471386968093}{927845029981902181528} a^{13} - \frac{2060957259615357644725}{231961257495475545382} a^{12} + \frac{5338969332289377406885}{463922514990951090764} a^{11} - \frac{1460372954756311061056}{115980628747737772691} a^{10} + \frac{3034046436742528462929}{231961257495475545382} a^{9} - \frac{4663002903479006791823}{463922514990951090764} a^{8} + \frac{1222719669959981957202}{115980628747737772691} a^{7} - \frac{2583753456758724502843}{463922514990951090764} a^{6} + \frac{3237440335185907297793}{463922514990951090764} a^{5} - \frac{867239400920002438461}{463922514990951090764} a^{4} + \frac{3017532106499784457889}{927845029981902181528} a^{3} - \frac{260523268758770467827}{231961257495475545382} a^{2} + \frac{241645083142882246752}{115980628747737772691} a + \frac{62820922550134507054}{115980628747737772691} \) (order $10$)
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:  \( 196321.686407 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T654:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 70 conjugacy class representatives for t20n654 are not computed
Character table for t20n654 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.6.189569215625.1

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 24 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
60662149Data not computed