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Properties

Label 20.12.2479222646...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{4}\cdot 5^{16}\cdot 6329^{5}$
Root discriminant $37.13$
Ramified primes $2, 5, 6329$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1037

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-284, -2668, -9224, -11516, 12221, 58405, 82962, 69751, 38592, 4741, -12048, -10905, -5014, -424, 918, 481, 91, -33, -22, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 22*x^18 - 33*x^17 + 91*x^16 + 481*x^15 + 918*x^14 - 424*x^13 - 5014*x^12 - 10905*x^11 - 12048*x^10 + 4741*x^9 + 38592*x^8 + 69751*x^7 + 82962*x^6 + 58405*x^5 + 12221*x^4 - 11516*x^3 - 9224*x^2 - 2668*x - 284)
 
gp: K = bnfinit(x^20 - 22*x^18 - 33*x^17 + 91*x^16 + 481*x^15 + 918*x^14 - 424*x^13 - 5014*x^12 - 10905*x^11 - 12048*x^10 + 4741*x^9 + 38592*x^8 + 69751*x^7 + 82962*x^6 + 58405*x^5 + 12221*x^4 - 11516*x^3 - 9224*x^2 - 2668*x - 284, 1)
 

Normalized defining polynomial

\( x^{20} - 22 x^{18} - 33 x^{17} + 91 x^{16} + 481 x^{15} + 918 x^{14} - 424 x^{13} - 5014 x^{12} - 10905 x^{11} - 12048 x^{10} + 4741 x^{9} + 38592 x^{8} + 69751 x^{7} + 82962 x^{6} + 58405 x^{5} + 12221 x^{4} - 11516 x^{3} - 9224 x^{2} - 2668 x - 284 \)

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:  \(24792226469882721311035156250000=2^{4}\cdot 5^{16}\cdot 6329^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.13$
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:  $2, 5, 6329$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{15} - \frac{2}{5} a^{14} - \frac{1}{5} a^{13} - \frac{1}{5} a^{12} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{17} + \frac{2}{5} a^{15} + \frac{2}{5} a^{14} - \frac{2}{5} a^{13} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{3} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{20} a^{18} - \frac{1}{10} a^{17} - \frac{1}{4} a^{15} - \frac{7}{20} a^{14} + \frac{1}{20} a^{13} - \frac{3}{10} a^{12} + \frac{3}{10} a^{11} - \frac{3}{10} a^{10} - \frac{1}{4} a^{9} - \frac{3}{10} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{20} a^{5} + \frac{2}{5} a^{4} - \frac{1}{4} a^{3} - \frac{1}{20} a^{2} + \frac{3}{10}$, $\frac{1}{9545859377750710852751671628686909880} a^{19} + \frac{9556072651524220456869806746668379}{2386464844437677713187917907171727470} a^{18} - \frac{78263648484896735984913395161992997}{2386464844437677713187917907171727470} a^{17} - \frac{644135798551074918332491570138771573}{9545859377750710852751671628686909880} a^{16} + \frac{3038490601229787817449263122599793847}{9545859377750710852751671628686909880} a^{15} + \frac{2926914623612495214189897580246113787}{9545859377750710852751671628686909880} a^{14} + \frac{3607172882377334879267051215415201}{2386464844437677713187917907171727470} a^{13} - \frac{1203391355854328076016330254161262347}{4772929688875355426375835814343454940} a^{12} + \frac{88342091323660689942425566679982315}{954585937775071085275167162868690988} a^{11} - \frac{577169725615561338262807345102180641}{1909171875550142170550334325737381976} a^{10} + \frac{573154307615865991053241497311416097}{1193232422218838856593958953585863735} a^{9} + \frac{2493361253249778410102155445573417411}{9545859377750710852751671628686909880} a^{8} + \frac{353180844214896358736405659091279348}{1193232422218838856593958953585863735} a^{7} - \frac{4368211310956064511206381250273492819}{9545859377750710852751671628686909880} a^{6} + \frac{1447658645818405842063551805095460441}{4772929688875355426375835814343454940} a^{5} + \frac{1802866365629248550273209316195185303}{9545859377750710852751671628686909880} a^{4} - \frac{2212595640938107635491804699136404439}{9545859377750710852751671628686909880} a^{3} + \frac{15232178519472978617523990032884341}{50241365146056372909219324361510052} a^{2} - \frac{103803598746180921043583645264800093}{4772929688875355426375835814343454940} a + \frac{664586881314877746421682109529280937}{2386464844437677713187917907171727470}$

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:  $15$
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:  \( 445897816.084 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.625878765625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ R $16{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ $16{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ $16{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
6329Data not computed