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Properties

Label 20.4.65861868411...0144.2
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 11^{16}\cdot 241^{3}$
Root discriminant $21.92$
Ramified primes $2, 11, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T747

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-43, -30, 210, -297, -32, 786, -1666, 2450, -3097, 3494, -3388, 2717, -1784, 937, -316, -38, 148, -109, 44, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 44*x^18 - 109*x^17 + 148*x^16 - 38*x^15 - 316*x^14 + 937*x^13 - 1784*x^12 + 2717*x^11 - 3388*x^10 + 3494*x^9 - 3097*x^8 + 2450*x^7 - 1666*x^6 + 786*x^5 - 32*x^4 - 297*x^3 + 210*x^2 - 30*x - 43)
 
gp: K = bnfinit(x^20 - 10*x^19 + 44*x^18 - 109*x^17 + 148*x^16 - 38*x^15 - 316*x^14 + 937*x^13 - 1784*x^12 + 2717*x^11 - 3388*x^10 + 3494*x^9 - 3097*x^8 + 2450*x^7 - 1666*x^6 + 786*x^5 - 32*x^4 - 297*x^3 + 210*x^2 - 30*x - 43, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 44 x^{18} - 109 x^{17} + 148 x^{16} - 38 x^{15} - 316 x^{14} + 937 x^{13} - 1784 x^{12} + 2717 x^{11} - 3388 x^{10} + 3494 x^{9} - 3097 x^{8} + 2450 x^{7} - 1666 x^{6} + 786 x^{5} - 32 x^{4} - 297 x^{3} + 210 x^{2} - 30 x - 43 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(658618684118710741619590144=2^{10}\cdot 11^{16}\cdot 241^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.92$
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:  $2, 11, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{4188092819868765616210601} a^{19} + \frac{1881076022333485099676130}{4188092819868765616210601} a^{18} + \frac{1852648731534103396285779}{4188092819868765616210601} a^{17} - \frac{991789169469754039027541}{4188092819868765616210601} a^{16} + \frac{928845304555643882998745}{4188092819868765616210601} a^{15} + \frac{1029533618411281092098004}{4188092819868765616210601} a^{14} + \frac{236989125917652715336494}{4188092819868765616210601} a^{13} + \frac{640942306823805556857176}{4188092819868765616210601} a^{12} + \frac{1397631728328808634714604}{4188092819868765616210601} a^{11} + \frac{111383551651622557487630}{4188092819868765616210601} a^{10} + \frac{21995784308230001662398}{4188092819868765616210601} a^{9} + \frac{293229666104138030924705}{4188092819868765616210601} a^{8} + \frac{726512028941106215801415}{4188092819868765616210601} a^{7} - \frac{654938745280524937836937}{4188092819868765616210601} a^{6} - \frac{104113084460143362615529}{4188092819868765616210601} a^{5} - \frac{1579076704130101328366662}{4188092819868765616210601} a^{4} - \frac{512799633903911632809712}{4188092819868765616210601} a^{3} + \frac{744006927146556206337284}{4188092819868765616210601} a^{2} - \frac{8156536377624550673887}{62508848057742770391203} a + \frac{1668098332948992291514876}{4188092819868765616210601}$

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:  $11$
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:  \( 245400.181357 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T747:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n747 are not computed
Character table for t20n747 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.51660490321.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ $20$ R $20$ $20$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
11Data not computed
241Data not computed