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Properties

Label 20.10.8351026207...0187.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,13^{10}\cdot 347^{7}$
Root discriminant $27.93$
Ramified primes $13, 347$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T796

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![53, -684, 1774, 881, -7057, 6629, -1628, -1785, 5682, -4221, -687, 230, 1645, -501, -536, 307, -42, -2, 7, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 7*x^18 - 2*x^17 - 42*x^16 + 307*x^15 - 536*x^14 - 501*x^13 + 1645*x^12 + 230*x^11 - 687*x^10 - 4221*x^9 + 5682*x^8 - 1785*x^7 - 1628*x^6 + 6629*x^5 - 7057*x^4 + 881*x^3 + 1774*x^2 - 684*x + 53)
 
gp: K = bnfinit(x^20 - 5*x^19 + 7*x^18 - 2*x^17 - 42*x^16 + 307*x^15 - 536*x^14 - 501*x^13 + 1645*x^12 + 230*x^11 - 687*x^10 - 4221*x^9 + 5682*x^8 - 1785*x^7 - 1628*x^6 + 6629*x^5 - 7057*x^4 + 881*x^3 + 1774*x^2 - 684*x + 53, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 7 x^{18} - 2 x^{17} - 42 x^{16} + 307 x^{15} - 536 x^{14} - 501 x^{13} + 1645 x^{12} + 230 x^{11} - 687 x^{10} - 4221 x^{9} + 5682 x^{8} - 1785 x^{7} - 1628 x^{6} + 6629 x^{5} - 7057 x^{4} + 881 x^{3} + 1774 x^{2} - 684 x + 53 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-83510262074750967216189850187=-\,13^{10}\cdot 347^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.93$
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:  $13, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{574689754261953626482668909859154077807} a^{19} - \frac{279050448729812060948569562082677273895}{574689754261953626482668909859154077807} a^{18} + \frac{48762231745320940582986845670417812062}{574689754261953626482668909859154077807} a^{17} - \frac{108544394788219377379463798169617157762}{574689754261953626482668909859154077807} a^{16} - \frac{163221215048372785530635272237928416792}{574689754261953626482668909859154077807} a^{15} + \frac{70533979285996325199622233085236305789}{574689754261953626482668909859154077807} a^{14} + \frac{113602731275163555673703900952601678606}{574689754261953626482668909859154077807} a^{13} - \frac{246652729554318641688787146284833633826}{574689754261953626482668909859154077807} a^{12} + \frac{196948835326616496558384634474221936818}{574689754261953626482668909859154077807} a^{11} + \frac{9660706135818910324524130250151576027}{574689754261953626482668909859154077807} a^{10} + \frac{135693780678968079343317869196750835735}{574689754261953626482668909859154077807} a^{9} - \frac{33851510671562749031598188190908142707}{574689754261953626482668909859154077807} a^{8} + \frac{143788144318744270373047713285693289814}{574689754261953626482668909859154077807} a^{7} + \frac{132001910861324909266189091071276322253}{574689754261953626482668909859154077807} a^{6} + \frac{133479748708791546348721777133163418221}{574689754261953626482668909859154077807} a^{5} - \frac{264969632073397616331653435796810332487}{574689754261953626482668909859154077807} a^{4} - \frac{46034872813253782329634309903037363446}{574689754261953626482668909859154077807} a^{3} + \frac{21281352539244843780907194718963212114}{574689754261953626482668909859154077807} a^{2} + \frac{2058859777291649085650684280326928974}{574689754261953626482668909859154077807} a + \frac{164970335204084145944339980786629493403}{574689754261953626482668909859154077807}$

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:  $14$
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:  \( 12429235.7338 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T796:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n796 are not computed
Character table for t20n796 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.3.4511.1, 10.6.44707018837.1

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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
347Data not computed