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Properties

Label 20.10.2205520391...7047.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,11^{18}\cdot 1583^{3}$
Root discriminant $26.13$
Ramified primes $11, 1583$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T432

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -1, -50, -128, 140, 789, 182, -1421, -638, 1013, 275, -413, 86, 138, -74, -24, 18, -4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 4*x^18 + 18*x^17 - 24*x^16 - 74*x^15 + 138*x^14 + 86*x^13 - 413*x^12 + 275*x^11 + 1013*x^10 - 638*x^9 - 1421*x^8 + 182*x^7 + 789*x^6 + 140*x^5 - 128*x^4 - 50*x^3 - x^2 + 4*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 - 4*x^18 + 18*x^17 - 24*x^16 - 74*x^15 + 138*x^14 + 86*x^13 - 413*x^12 + 275*x^11 + 1013*x^10 - 638*x^9 - 1421*x^8 + 182*x^7 + 789*x^6 + 140*x^5 - 128*x^4 - 50*x^3 - x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 4 x^{18} + 18 x^{17} - 24 x^{16} - 74 x^{15} + 138 x^{14} + 86 x^{13} - 413 x^{12} + 275 x^{11} + 1013 x^{10} - 638 x^{9} - 1421 x^{8} + 182 x^{7} + 789 x^{6} + 140 x^{5} - 128 x^{4} - 50 x^{3} - x^{2} + 4 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-22055203913038149640193817047=-\,11^{18}\cdot 1583^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.13$
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:  $11, 1583$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{39990955817429955073} a^{19} - \frac{17631022068248889638}{39990955817429955073} a^{18} + \frac{19609131303338589058}{39990955817429955073} a^{17} + \frac{9912850194836773321}{39990955817429955073} a^{16} - \frac{8358026841298550711}{39990955817429955073} a^{15} + \frac{18405730766269451743}{39990955817429955073} a^{14} - \frac{14445800081381218063}{39990955817429955073} a^{13} - \frac{9656941191377621261}{39990955817429955073} a^{12} - \frac{12113839220862520021}{39990955817429955073} a^{11} + \frac{13022018333185941965}{39990955817429955073} a^{10} - \frac{6072880368562987386}{39990955817429955073} a^{9} + \frac{15072877469828728586}{39990955817429955073} a^{8} + \frac{13719348023022808083}{39990955817429955073} a^{7} - \frac{7347594761198717047}{39990955817429955073} a^{6} + \frac{14227459680658565496}{39990955817429955073} a^{5} + \frac{19795909871626061361}{39990955817429955073} a^{4} - \frac{8389581353005241344}{39990955817429955073} a^{3} - \frac{8966304414834830682}{39990955817429955073} a^{2} + \frac{9671661916824108637}{39990955817429955073} a + \frac{8206536532705618161}{39990955817429955073}$

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:  \( 5953104.25923 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T432:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n432 are not computed
Character table for t20n432 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.3732631194853.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
1583Data not computed