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Properties

Label 20.12.5834302965...2368.6
Degree $20$
Signature $[12, 4]$
Discriminant $2^{10}\cdot 19^{10}\cdot 43^{11}$
Root discriminant $48.79$
Ramified primes $2, 19, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T423

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, -567, -2767, 5123, 6088, -6958, -6228, -1006, 8206, -462, -2220, 2179, -44, -2097, 381, 399, -80, 13, -5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 5*x^18 + 13*x^17 - 80*x^16 + 399*x^15 + 381*x^14 - 2097*x^13 - 44*x^12 + 2179*x^11 - 2220*x^10 - 462*x^9 + 8206*x^8 - 1006*x^7 - 6228*x^6 - 6958*x^5 + 6088*x^4 + 5123*x^3 - 2767*x^2 - 567*x + 49)
 
gp: K = bnfinit(x^20 - 4*x^19 - 5*x^18 + 13*x^17 - 80*x^16 + 399*x^15 + 381*x^14 - 2097*x^13 - 44*x^12 + 2179*x^11 - 2220*x^10 - 462*x^9 + 8206*x^8 - 1006*x^7 - 6228*x^6 - 6958*x^5 + 6088*x^4 + 5123*x^3 - 2767*x^2 - 567*x + 49, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 5 x^{18} + 13 x^{17} - 80 x^{16} + 399 x^{15} + 381 x^{14} - 2097 x^{13} - 44 x^{12} + 2179 x^{11} - 2220 x^{10} - 462 x^{9} + 8206 x^{8} - 1006 x^{7} - 6228 x^{6} - 6958 x^{5} + 6088 x^{4} + 5123 x^{3} - 2767 x^{2} - 567 x + 49 \)

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:  \(5834302965409512291058019417402368=2^{10}\cdot 19^{10}\cdot 43^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.79$
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:  $2, 19, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{10} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{4} + \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{25} a^{14} + \frac{2}{25} a^{13} + \frac{2}{25} a^{12} + \frac{2}{25} a^{11} - \frac{1}{5} a^{10} + \frac{8}{25} a^{9} + \frac{11}{25} a^{8} + \frac{8}{25} a^{7} - \frac{9}{25} a^{6} + \frac{11}{25} a^{5} + \frac{12}{25} a^{4} + \frac{9}{25} a^{3} - \frac{2}{25} a^{2} - \frac{4}{25}$, $\frac{1}{25} a^{15} - \frac{2}{25} a^{13} - \frac{2}{25} a^{12} + \frac{1}{25} a^{11} - \frac{7}{25} a^{10} + \frac{2}{5} a^{9} - \frac{4}{25} a^{8} + \frac{2}{5} a^{7} - \frac{11}{25} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{4}{25} a^{2} + \frac{1}{25} a + \frac{3}{25}$, $\frac{1}{25} a^{16} + \frac{2}{25} a^{13} + \frac{2}{25} a^{11} + \frac{1}{5} a^{10} + \frac{2}{25} a^{9} + \frac{7}{25} a^{8} + \frac{1}{5} a^{7} - \frac{8}{25} a^{6} + \frac{2}{25} a^{5} - \frac{6}{25} a^{4} + \frac{7}{25} a^{3} + \frac{7}{25} a^{2} + \frac{8}{25} a + \frac{2}{25}$, $\frac{1}{125} a^{17} - \frac{1}{125} a^{16} - \frac{1}{125} a^{15} - \frac{9}{125} a^{13} - \frac{7}{125} a^{11} + \frac{34}{125} a^{10} - \frac{16}{125} a^{9} + \frac{3}{25} a^{8} - \frac{14}{125} a^{7} - \frac{51}{125} a^{6} - \frac{3}{25} a^{5} + \frac{34}{125} a^{4} - \frac{3}{125} a^{3} + \frac{16}{125} a^{2} + \frac{38}{125} a - \frac{42}{125}$, $\frac{1}{125} a^{18} - \frac{2}{125} a^{16} - \frac{1}{125} a^{15} + \frac{1}{125} a^{14} + \frac{11}{125} a^{13} - \frac{12}{125} a^{12} - \frac{3}{125} a^{11} - \frac{7}{125} a^{10} - \frac{21}{125} a^{9} + \frac{36}{125} a^{8} - \frac{12}{25} a^{7} - \frac{31}{125} a^{6} - \frac{46}{125} a^{5} + \frac{51}{125} a^{4} + \frac{3}{125} a^{3} - \frac{41}{125} a^{2} + \frac{46}{125} a - \frac{57}{125}$, $\frac{1}{4256706681493053686124128875} a^{19} - \frac{11035329127519594153080367}{4256706681493053686124128875} a^{18} + \frac{638864635335726596736419}{170268267259722147444965155} a^{17} - \frac{69533929556568301841722994}{4256706681493053686124128875} a^{16} - \frac{32848374992058205595118974}{4256706681493053686124128875} a^{15} + \frac{6328222809423410774649777}{608100954499007669446304125} a^{14} + \frac{163340280938940455685016328}{4256706681493053686124128875} a^{13} + \frac{2161840719820323335756721}{4256706681493053686124128875} a^{12} + \frac{66820821652506363654101611}{851341336298610737224825775} a^{11} + \frac{936287838552192938701399071}{4256706681493053686124128875} a^{10} + \frac{1283084354245404534821362371}{4256706681493053686124128875} a^{9} + \frac{15328484744553777362777409}{608100954499007669446304125} a^{8} + \frac{1411934971275672442712857746}{4256706681493053686124128875} a^{7} + \frac{2038977485646015651168469389}{4256706681493053686124128875} a^{6} + \frac{1122991931569700827741881448}{4256706681493053686124128875} a^{5} - \frac{238206778846923086259538833}{608100954499007669446304125} a^{4} - \frac{1715628438271688097325604018}{4256706681493053686124128875} a^{3} - \frac{14276373604767560171420401}{34053653451944429488993031} a^{2} + \frac{1881161236471668594746221847}{4256706681493053686124128875} a - \frac{20478079203047265370272419}{121620190899801533889260825}$

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:  \( 6816927907.74 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T423:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 160 conjugacy class representatives for t20n423 are not computed
Character table for t20n423 is not computed

Intermediate fields

\(\Q(\sqrt{817}) \), 5.5.667489.1 x5, 10.10.364007458703857.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.14$x^{10} + 5 x^{8} - 50 x^{6} - 58 x^{4} + 49 x^{2} + 21$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.4.3.1$x^{4} + 387$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$