↑

Properties

Label 20.6.22728270974...6875.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,5^{10}\cdot 11^{11}\cdot 13^{8}$
Root discriminant $23.33$
Ramified primes $5, 11, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T547

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 8, 25, -2, -54, 4, -3, -29, -28, -20, 21, -29, 59, -1, 3, -4, -12, 4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 4*x^18 - 12*x^17 - 4*x^16 + 3*x^15 - x^14 + 59*x^13 - 29*x^12 + 21*x^11 - 20*x^10 - 28*x^9 - 29*x^8 - 3*x^7 + 4*x^6 - 54*x^5 - 2*x^4 + 25*x^3 + 8*x^2 - 1)
 
gp: K = bnfinit(x^20 - x^19 + 4*x^18 - 12*x^17 - 4*x^16 + 3*x^15 - x^14 + 59*x^13 - 29*x^12 + 21*x^11 - 20*x^10 - 28*x^9 - 29*x^8 - 3*x^7 + 4*x^6 - 54*x^5 - 2*x^4 + 25*x^3 + 8*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 4 x^{18} - 12 x^{17} - 4 x^{16} + 3 x^{15} - x^{14} + 59 x^{13} - 29 x^{12} + 21 x^{11} - 20 x^{10} - 28 x^{9} - 29 x^{8} - 3 x^{7} + 4 x^{6} - 54 x^{5} - 2 x^{4} + 25 x^{3} + 8 x^{2} - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2272827097433843169248046875=-\,5^{10}\cdot 11^{11}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.33$
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:  $5, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11} a^{18} + \frac{5}{11} a^{17} + \frac{3}{11} a^{16} + \frac{5}{11} a^{15} - \frac{1}{11} a^{14} - \frac{4}{11} a^{13} - \frac{5}{11} a^{12} - \frac{1}{11} a^{11} - \frac{1}{11} a^{10} + \frac{2}{11} a^{9} + \frac{1}{11} a^{8} + \frac{4}{11} a^{7} - \frac{3}{11} a^{6} - \frac{2}{11} a^{5} - \frac{3}{11} a^{4} + \frac{1}{11} a^{3} - \frac{2}{11} a^{2} + \frac{4}{11} a - \frac{5}{11}$, $\frac{1}{180764213786763959} a^{19} - \frac{1286375423802871}{180764213786763959} a^{18} + \frac{47606434529584615}{180764213786763959} a^{17} - \frac{70578161371168963}{180764213786763959} a^{16} + \frac{20287221556465731}{180764213786763959} a^{15} + \frac{35346086329524901}{180764213786763959} a^{14} - \frac{86178215746647084}{180764213786763959} a^{13} - \frac{66366692311842132}{180764213786763959} a^{12} - \frac{65217966380657740}{180764213786763959} a^{11} - \frac{66554893386122887}{180764213786763959} a^{10} - \frac{60185701578172972}{180764213786763959} a^{9} + \frac{85888957867330589}{180764213786763959} a^{8} - \frac{50102743348279636}{180764213786763959} a^{7} - \frac{16551034332425947}{180764213786763959} a^{6} + \frac{26517876479268512}{180764213786763959} a^{5} + \frac{58854911097608055}{180764213786763959} a^{4} - \frac{35939619642703636}{180764213786763959} a^{3} + \frac{10658977985862265}{180764213786763959} a^{2} - \frac{25147170827260773}{180764213786763959} a + \frac{4250600927623328}{16433110344251269}$

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:  $12$
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:  \( 709138.335557 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T547:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 28800
The 41 conjugacy class representatives for t20n547
Character table for t20n547 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 10.6.1306755003125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: 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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ R R ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.12.8.1$x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$