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Properties

Label 20.12.1713729462...8125.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{13}\cdot 97^{2}\cdot 419^{4}\cdot 695771^{2}$
Root discriminant $57.77$
Ramified primes $5, 97, 419, 695771$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1039

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21731, -366925, 406400, 1875378, -821616, -547740, 408264, -305712, 60948, 14499, -62649, 31035, -4153, -755, 1411, -923, 159, 101, -30, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 30*x^18 + 101*x^17 + 159*x^16 - 923*x^15 + 1411*x^14 - 755*x^13 - 4153*x^12 + 31035*x^11 - 62649*x^10 + 14499*x^9 + 60948*x^8 - 305712*x^7 + 408264*x^6 - 547740*x^5 - 821616*x^4 + 1875378*x^3 + 406400*x^2 - 366925*x + 21731)
 
gp: K = bnfinit(x^20 - 3*x^19 - 30*x^18 + 101*x^17 + 159*x^16 - 923*x^15 + 1411*x^14 - 755*x^13 - 4153*x^12 + 31035*x^11 - 62649*x^10 + 14499*x^9 + 60948*x^8 - 305712*x^7 + 408264*x^6 - 547740*x^5 - 821616*x^4 + 1875378*x^3 + 406400*x^2 - 366925*x + 21731, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 30 x^{18} + 101 x^{17} + 159 x^{16} - 923 x^{15} + 1411 x^{14} - 755 x^{13} - 4153 x^{12} + 31035 x^{11} - 62649 x^{10} + 14499 x^{9} + 60948 x^{8} - 305712 x^{7} + 408264 x^{6} - 547740 x^{5} - 821616 x^{4} + 1875378 x^{3} + 406400 x^{2} - 366925 x + 21731 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(171372946259251659623031065673828125=5^{13}\cdot 97^{2}\cdot 419^{4}\cdot 695771^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.77$
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:  $5, 97, 419, 695771$
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}{21465882108675771423760500830517277487488695941785575113374765939} a^{19} + \frac{383802005454705169976718485226076236626116425919266657259217840}{3066554586953681631965785832931039641069813705969367873339252277} a^{18} + \frac{1871577432249531054049406474241686471060926691345666774563364300}{21465882108675771423760500830517277487488695941785575113374765939} a^{17} - \frac{576865902004178026986319488395011177776327115153877079275589511}{21465882108675771423760500830517277487488695941785575113374765939} a^{16} + \frac{6166574979457428294735039379574926779066666184931851762066432772}{21465882108675771423760500830517277487488695941785575113374765939} a^{15} - \frac{9748591958685806821966472978074713609631007076172250274525251146}{21465882108675771423760500830517277487488695941785575113374765939} a^{14} - \frac{10086398648720583807072934183828331403088639552334895120534009825}{21465882108675771423760500830517277487488695941785575113374765939} a^{13} - \frac{2915959858675708004178184179929184098179526807568820630332776186}{21465882108675771423760500830517277487488695941785575113374765939} a^{12} + \frac{423640757938489151948605855646790798851593079653386120059995634}{21465882108675771423760500830517277487488695941785575113374765939} a^{11} - \frac{6166379698779332362761964509298269023728851524784523975144829580}{21465882108675771423760500830517277487488695941785575113374765939} a^{10} - \frac{10574305035331329756296859238715466745814665387710141160971865641}{21465882108675771423760500830517277487488695941785575113374765939} a^{9} - \frac{470659931884849773018168027479092522899211586929030469469250232}{3066554586953681631965785832931039641069813705969367873339252277} a^{8} + \frac{1696129007628749755869708513768149377327854689947600615151491416}{21465882108675771423760500830517277487488695941785575113374765939} a^{7} + \frac{296763773358747513921820612471980730841037956691380734534739870}{21465882108675771423760500830517277487488695941785575113374765939} a^{6} - \frac{3524492953831725962173386029711186905349187462878109606793779227}{21465882108675771423760500830517277487488695941785575113374765939} a^{5} - \frac{1266345213246864643472148090899340504249284606623863694960893171}{21465882108675771423760500830517277487488695941785575113374765939} a^{4} + \frac{714048953396948798812213501227152084426043222593527602845310850}{3066554586953681631965785832931039641069813705969367873339252277} a^{3} + \frac{6103430556271600507082071262392363933949104740679665731601368300}{21465882108675771423760500830517277487488695941785575113374765939} a^{2} - \frac{8285554739170648632544592037946595517950262691416985258868059077}{21465882108675771423760500830517277487488695941785575113374765939} a - \frac{7475052636753472194863849340827719144363642582265318469135804265}{21465882108675771423760500830517277487488695941785575113374765939}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  $15$
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:  \( 11078408249.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1039:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 378 conjugacy class representatives for t20n1039 are not computed
Character table for t20n1039 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.911025153125.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
97Data not computed
419Data not computed
695771Data not computed