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Properties

Label 20.20.8169342613...7408.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{10}\cdot 3^{15}\cdot 11^{18}$
Root discriminant $27.90$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times D_4$ (as 20T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, -150, 641, -190, -3851, 6273, 3875, -14821, 3921, 12429, -7485, -4382, 4100, 498, -1022, 63, 118, -18, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 18*x^18 + 118*x^17 + 63*x^16 - 1022*x^15 + 498*x^14 + 4100*x^13 - 4382*x^12 - 7485*x^11 + 12429*x^10 + 3921*x^9 - 14821*x^8 + 3875*x^7 + 6273*x^6 - 3851*x^5 - 190*x^4 + 641*x^3 - 150*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 - 18*x^18 + 118*x^17 + 63*x^16 - 1022*x^15 + 498*x^14 + 4100*x^13 - 4382*x^12 - 7485*x^11 + 12429*x^10 + 3921*x^9 - 14821*x^8 + 3875*x^7 + 6273*x^6 - 3851*x^5 - 190*x^4 + 641*x^3 - 150*x^2 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 18 x^{18} + 118 x^{17} + 63 x^{16} - 1022 x^{15} + 498 x^{14} + 4100 x^{13} - 4382 x^{12} - 7485 x^{11} + 12429 x^{10} + 3921 x^{9} - 14821 x^{8} + 3875 x^{7} + 6273 x^{6} - 3851 x^{5} - 190 x^{4} + 641 x^{3} - 150 x^{2} + 5 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(81693426134005631737181457408=2^{10}\cdot 3^{15}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.90$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{67} a^{18} + \frac{23}{67} a^{17} - \frac{17}{67} a^{16} - \frac{5}{67} a^{15} - \frac{18}{67} a^{13} - \frac{6}{67} a^{12} + \frac{29}{67} a^{11} + \frac{20}{67} a^{10} + \frac{22}{67} a^{9} - \frac{16}{67} a^{8} - \frac{20}{67} a^{7} - \frac{1}{67} a^{6} + \frac{24}{67} a^{5} + \frac{17}{67} a^{4} + \frac{20}{67} a^{3} + \frac{25}{67} a^{2} + \frac{5}{67} a - \frac{5}{67}$, $\frac{1}{662430660027704501} a^{19} - \frac{127525298418945}{662430660027704501} a^{18} - \frac{186933545942170248}{662430660027704501} a^{17} - \frac{282731975670641099}{662430660027704501} a^{16} + \frac{269404157704395606}{662430660027704501} a^{15} + \frac{169011545207129654}{662430660027704501} a^{14} - \frac{944946081856582}{9887024776532903} a^{13} - \frac{115445837941008889}{662430660027704501} a^{12} - \frac{173915351041778628}{662430660027704501} a^{11} + \frac{312995327110735154}{662430660027704501} a^{10} + \frac{177377763467216480}{662430660027704501} a^{9} - \frac{61592621777296942}{662430660027704501} a^{8} + \frac{55201946406839386}{662430660027704501} a^{7} - \frac{47859021589800280}{662430660027704501} a^{6} + \frac{240732514522789661}{662430660027704501} a^{5} + \frac{42196304900447347}{662430660027704501} a^{4} + \frac{317860045804410506}{662430660027704501} a^{3} - \frac{185851441784824183}{662430660027704501} a^{2} + \frac{102638514673600144}{662430660027704501} a + \frac{193009925709159223}{662430660027704501}$

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:  $19$
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:  \( 71784662.2779 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times D_4$ (as 20T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 25 conjugacy class representatives for $C_5\times D_4$
Character table for $C_5\times D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{33}) \), 4.4.13068.1, \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{33})^+\)

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

Sibling fields

Galois closure: data not computed
Degree 20 sibling: 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 $20$ $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
3Data not computed
11Data not computed