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Properties

Label 20.6.12643669953...9375.2
Degree $20$
Signature $[6, 7]$
Discriminant $-\,5^{10}\cdot 1039^{3}\cdot 1049^{3}$
Root discriminant $17.99$
Ramified primes $5, 1039, 1049$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -33, 144, -337, 406, -89, -706, 1334, -1079, 141, 593, -778, 596, -316, 108, -1, -26, 17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 17*x^18 - 26*x^17 - x^16 + 108*x^15 - 316*x^14 + 596*x^13 - 778*x^12 + 593*x^11 + 141*x^10 - 1079*x^9 + 1334*x^8 - 706*x^7 - 89*x^6 + 406*x^5 - 337*x^4 + 144*x^3 - 33*x^2 - x + 1)
 
gp: K = bnfinit(x^20 - 6*x^19 + 17*x^18 - 26*x^17 - x^16 + 108*x^15 - 316*x^14 + 596*x^13 - 778*x^12 + 593*x^11 + 141*x^10 - 1079*x^9 + 1334*x^8 - 706*x^7 - 89*x^6 + 406*x^5 - 337*x^4 + 144*x^3 - 33*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 17 x^{18} - 26 x^{17} - x^{16} + 108 x^{15} - 316 x^{14} + 596 x^{13} - 778 x^{12} + 593 x^{11} + 141 x^{10} - 1079 x^{9} + 1334 x^{8} - 706 x^{7} - 89 x^{6} + 406 x^{5} - 337 x^{4} + 144 x^{3} - 33 x^{2} - x + 1 \)

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:  \(-12643669953134424130859375=-\,5^{10}\cdot 1039^{3}\cdot 1049^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.99$
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:  $5, 1039, 1049$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{17298932656988558798267} a^{19} - \frac{6484852445981061718653}{17298932656988558798267} a^{18} - \frac{5035400252357972506724}{17298932656988558798267} a^{17} + \frac{2696072315212026677980}{17298932656988558798267} a^{16} - \frac{2950796833771884530120}{17298932656988558798267} a^{15} + \frac{7282683875905373740672}{17298932656988558798267} a^{14} + \frac{5035689905207734312161}{17298932656988558798267} a^{13} + \frac{6471313536655898981809}{17298932656988558798267} a^{12} - \frac{1089657423954043130718}{17298932656988558798267} a^{11} + \frac{1664907209040621059805}{17298932656988558798267} a^{10} + \frac{462012346329212515353}{17298932656988558798267} a^{9} - \frac{7768156935935939270997}{17298932656988558798267} a^{8} - \frac{4460802817115431518765}{17298932656988558798267} a^{7} + \frac{8476067214702182270575}{17298932656988558798267} a^{6} + \frac{2301700075625482834841}{17298932656988558798267} a^{5} - \frac{5673456511874624977625}{17298932656988558798267} a^{4} + \frac{3327070303324758373767}{17298932656988558798267} a^{3} - \frac{2697532762569399744321}{17298932656988558798267} a^{2} - \frac{6813282182779942437235}{17298932656988558798267} a + \frac{2694803189506236931100}{17298932656988558798267}$

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:  \( 49260.6111116 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1022 are not computed
Character table for t20n1022 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.4.3405971875.1

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 32 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
1039Data not computed
1049Data not computed