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Properties

Label 20.0.15802861880...9424.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 761^{5}$
Root discriminant $18.19$
Ramified primes $2, 3, 761$
Class number $2$
Class group $[2]$
Galois group $C_2\times D_5\wr C_2$ (as 20T92)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![153, -1296, 4962, -11076, 15361, -12624, 4435, 1604, -1680, -862, 1928, -1288, 456, -36, -62, 20, 15, -16, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 10*x^18 - 16*x^17 + 15*x^16 + 20*x^15 - 62*x^14 - 36*x^13 + 456*x^12 - 1288*x^11 + 1928*x^10 - 862*x^9 - 1680*x^8 + 1604*x^7 + 4435*x^6 - 12624*x^5 + 15361*x^4 - 11076*x^3 + 4962*x^2 - 1296*x + 153)
 
gp: K = bnfinit(x^20 - 4*x^19 + 10*x^18 - 16*x^17 + 15*x^16 + 20*x^15 - 62*x^14 - 36*x^13 + 456*x^12 - 1288*x^11 + 1928*x^10 - 862*x^9 - 1680*x^8 + 1604*x^7 + 4435*x^6 - 12624*x^5 + 15361*x^4 - 11076*x^3 + 4962*x^2 - 1296*x + 153, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 10 x^{18} - 16 x^{17} + 15 x^{16} + 20 x^{15} - 62 x^{14} - 36 x^{13} + 456 x^{12} - 1288 x^{11} + 1928 x^{10} - 862 x^{9} - 1680 x^{8} + 1604 x^{7} + 4435 x^{6} - 12624 x^{5} + 15361 x^{4} - 11076 x^{3} + 4962 x^{2} - 1296 x + 153 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15802861880872333999079424=2^{20}\cdot 3^{10}\cdot 761^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.19$
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:  $2, 3, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{51} a^{18} + \frac{5}{51} a^{17} - \frac{8}{51} a^{16} + \frac{5}{51} a^{15} + \frac{1}{17} a^{14} - \frac{13}{51} a^{13} - \frac{11}{51} a^{12} + \frac{7}{17} a^{11} + \frac{4}{17} a^{10} - \frac{4}{51} a^{9} + \frac{14}{51} a^{8} - \frac{25}{51} a^{7} + \frac{6}{17} a^{6} - \frac{25}{51} a^{5} + \frac{16}{51} a^{4} + \frac{3}{17} a^{3} + \frac{1}{51} a^{2} - \frac{2}{17} a$, $\frac{1}{98146670950694359384727130903} a^{19} - \frac{472886257922315101360805944}{98146670950694359384727130903} a^{18} - \frac{7791286532705596091292640232}{98146670950694359384727130903} a^{17} - \frac{5409596316351102816535678672}{98146670950694359384727130903} a^{16} - \frac{1158730845972449934473137665}{32715556983564786461575710301} a^{15} - \frac{39988304407605799550516992918}{98146670950694359384727130903} a^{14} + \frac{20450622681427189655399640706}{98146670950694359384727130903} a^{13} - \frac{3553078216592720962622668632}{32715556983564786461575710301} a^{12} - \frac{8985558291142549393294550571}{32715556983564786461575710301} a^{11} - \frac{5365280853035109988081837264}{98146670950694359384727130903} a^{10} + \frac{29544301380941031029465211557}{98146670950694359384727130903} a^{9} - \frac{29248944092534744076292387579}{98146670950694359384727130903} a^{8} + \frac{2710684623482464288663222920}{32715556983564786461575710301} a^{7} - \frac{41145157276123266483717177403}{98146670950694359384727130903} a^{6} - \frac{17038424169625790212054329758}{98146670950694359384727130903} a^{5} + \frac{13346871209067140333137252238}{32715556983564786461575710301} a^{4} - \frac{16028989175419550157480652517}{98146670950694359384727130903} a^{3} + \frac{2961834141763700249493985588}{32715556983564786461575710301} a^{2} - \frac{13851752959118014178897376181}{32715556983564786461575710301} a - \frac{822580040196332621978700746}{1924444528444987438916218253}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{310010460051756382}{8035420355712239631} a^{19} + \frac{528305875585365050}{2678473451904079877} a^{18} - \frac{1287042409346182259}{2678473451904079877} a^{17} + \frac{2211708349597467011}{2678473451904079877} a^{16} - \frac{6068683633580523242}{8035420355712239631} a^{15} - \frac{6068432282182022462}{8035420355712239631} a^{14} + \frac{9729087271808434070}{2678473451904079877} a^{13} + \frac{6213921414283335881}{8035420355712239631} a^{12} - \frac{57956626086497645670}{2678473451904079877} a^{11} + \frac{509474895981708849121}{8035420355712239631} a^{10} - \frac{810852284919017472361}{8035420355712239631} a^{9} + \frac{402973701678471034766}{8035420355712239631} a^{8} + \frac{788524081710507245059}{8035420355712239631} a^{7} - \frac{905764331326896226097}{8035420355712239631} a^{6} - \frac{33981860663491930022}{157557261876710581} a^{5} + \frac{5379180966729819147674}{8035420355712239631} a^{4} - \frac{6333860532041449598374}{8035420355712239631} a^{3} + \frac{4084604868865271927834}{8035420355712239631} a^{2} - \frac{488824658118209662279}{2678473451904079877} a + \frac{4593170470888838189}{157557261876710581} \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 21318.5277295 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_5\wr C_2$ (as 20T92):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $C_2\times D_5\wr C_2$
Character table for $C_2\times D_5\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.109584.1, 10.0.593019904.1

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 R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
2Data not computed
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$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}$
761Data not computed