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Properties

Label 20.14.4075596666...9375.2
Degree $20$
Signature $[14, 3]$
Discriminant $-\,3^{8}\cdot 5^{12}\cdot 239^{9}$
Root discriminant $47.92$
Ramified primes $3, 5, 239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T525

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-25541, -125501, -64330, 286818, 101822, -313055, 82548, 121493, -148264, 55042, 24582, -33361, 14905, -1513, -2187, 1276, -287, -12, 33, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 33*x^18 - 12*x^17 - 287*x^16 + 1276*x^15 - 2187*x^14 - 1513*x^13 + 14905*x^12 - 33361*x^11 + 24582*x^10 + 55042*x^9 - 148264*x^8 + 121493*x^7 + 82548*x^6 - 313055*x^5 + 101822*x^4 + 286818*x^3 - 64330*x^2 - 125501*x - 25541)
 
gp: K = bnfinit(x^20 - 10*x^19 + 33*x^18 - 12*x^17 - 287*x^16 + 1276*x^15 - 2187*x^14 - 1513*x^13 + 14905*x^12 - 33361*x^11 + 24582*x^10 + 55042*x^9 - 148264*x^8 + 121493*x^7 + 82548*x^6 - 313055*x^5 + 101822*x^4 + 286818*x^3 - 64330*x^2 - 125501*x - 25541, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 33 x^{18} - 12 x^{17} - 287 x^{16} + 1276 x^{15} - 2187 x^{14} - 1513 x^{13} + 14905 x^{12} - 33361 x^{11} + 24582 x^{10} + 55042 x^{9} - 148264 x^{8} + 121493 x^{7} + 82548 x^{6} - 313055 x^{5} + 101822 x^{4} + 286818 x^{3} - 64330 x^{2} - 125501 x - 25541 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-4075596666226495465452050537109375=-\,3^{8}\cdot 5^{12}\cdot 239^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.92$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} - \frac{2}{11} a^{14} - \frac{3}{11} a^{13} + \frac{4}{11} a^{12} - \frac{5}{11} a^{9} - \frac{5}{11} a^{8} + \frac{4}{11} a^{7} + \frac{2}{11} a^{6} + \frac{3}{11} a^{5} + \frac{1}{11} a^{4} + \frac{4}{11} a^{2} - \frac{1}{11} a + \frac{4}{11}$, $\frac{1}{3905} a^{16} - \frac{8}{3905} a^{15} + \frac{1373}{3905} a^{14} - \frac{151}{355} a^{13} - \frac{266}{3905} a^{12} - \frac{11}{71} a^{11} - \frac{874}{3905} a^{10} - \frac{1383}{3905} a^{9} - \frac{1869}{3905} a^{8} - \frac{4}{355} a^{7} - \frac{290}{781} a^{6} - \frac{1216}{3905} a^{5} - \frac{1084}{3905} a^{4} - \frac{1052}{3905} a^{3} + \frac{1537}{3905} a^{2} + \frac{791}{3905} a + \frac{1274}{3905}$, $\frac{1}{113245} a^{17} + \frac{6}{113245} a^{16} + \frac{4811}{113245} a^{15} - \frac{1534}{10295} a^{14} - \frac{7615}{22649} a^{13} + \frac{21586}{113245} a^{12} + \frac{10181}{113245} a^{11} - \frac{37049}{113245} a^{10} - \frac{54601}{113245} a^{9} - \frac{8011}{22649} a^{8} + \frac{8229}{113245} a^{7} - \frac{10511}{113245} a^{6} + \frac{3227}{10295} a^{5} - \frac{20488}{113245} a^{4} - \frac{17096}{113245} a^{3} + \frac{2254}{10295} a^{2} + \frac{51753}{113245} a + \frac{796}{113245}$, $\frac{1}{218352525993159855} a^{18} - \frac{3}{72784175331053285} a^{17} - \frac{18086605126946}{218352525993159855} a^{16} + \frac{144692841015772}{218352525993159855} a^{15} + \frac{600595947811347}{2509799149346665} a^{14} + \frac{68409995051432663}{218352525993159855} a^{13} - \frac{71755534399391377}{218352525993159855} a^{12} - \frac{5169384339912123}{72784175331053285} a^{11} - \frac{28249808707994011}{72784175331053285} a^{10} + \frac{68359767205039961}{218352525993159855} a^{9} + \frac{12903795847798232}{218352525993159855} a^{8} + \frac{34890183779055377}{218352525993159855} a^{7} + \frac{14634529040139214}{72784175331053285} a^{6} - \frac{1328651966752474}{7529397448039995} a^{5} + \frac{3523486333917757}{218352525993159855} a^{4} + \frac{3853711380418946}{72784175331053285} a^{3} + \frac{1068309469755853}{72784175331053285} a^{2} - \frac{29530437001576837}{72784175331053285} a - \frac{103867453773365218}{218352525993159855}$, $\frac{1}{2401877785924758405} a^{19} - \frac{4}{2401877785924758405} a^{18} + \frac{1194822396799}{2401877785924758405} a^{17} - \frac{36544106190391}{800625928641586135} a^{16} + \frac{88424216167153964}{2401877785924758405} a^{15} + \frac{28666004559855622}{82823371928439945} a^{14} + \frac{132514073933674681}{800625928641586135} a^{13} - \frac{776968798137285509}{2401877785924758405} a^{12} - \frac{296200761538769796}{800625928641586135} a^{11} + \frac{4365333021624016}{218352525993159855} a^{10} + \frac{213796179003489884}{800625928641586135} a^{9} - \frac{314793758422854406}{800625928641586135} a^{8} - \frac{940642461725560478}{2401877785924758405} a^{7} - \frac{172094472249376016}{2401877785924758405} a^{6} - \frac{3068053092086326}{11276421530163185} a^{5} + \frac{96007993608301763}{2401877785924758405} a^{4} - \frac{19077677829174212}{72784175331053285} a^{3} + \frac{7022860447140787}{27607790642813315} a^{2} + \frac{85864331249480152}{218352525993159855} a - \frac{219675563766752716}{480375557184951681}$

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:  $16$
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:  \( 11158823549.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
239Data not computed