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Properties

Label 20.8.49338146756...1664.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{30}\cdot 11^{16}$
Root discriminant $19.26$
Ramified primes $2, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4:C_5$ (as 20T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -34, 20, 152, -68, -196, 68, 180, -70, -330, 320, 201, -428, 140, 134, -138, 40, 6, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 6*x^18 + 40*x^17 - 138*x^16 + 134*x^15 + 140*x^14 - 428*x^13 + 201*x^12 + 320*x^11 - 330*x^10 - 70*x^9 + 180*x^8 + 68*x^7 - 196*x^6 - 68*x^5 + 152*x^4 + 20*x^3 - 34*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^20 - 6*x^19 + 6*x^18 + 40*x^17 - 138*x^16 + 134*x^15 + 140*x^14 - 428*x^13 + 201*x^12 + 320*x^11 - 330*x^10 - 70*x^9 + 180*x^8 + 68*x^7 - 196*x^6 - 68*x^5 + 152*x^4 + 20*x^3 - 34*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 6 x^{18} + 40 x^{17} - 138 x^{16} + 134 x^{15} + 140 x^{14} - 428 x^{13} + 201 x^{12} + 320 x^{11} - 330 x^{10} - 70 x^{9} + 180 x^{8} + 68 x^{7} - 196 x^{6} - 68 x^{5} + 152 x^{4} + 20 x^{3} - 34 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(49338146756019243307761664=2^{30}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.26$
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:  $2, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{646350777554489400361} a^{19} + \frac{250629404825673452611}{646350777554489400361} a^{18} + \frac{300539439810250699047}{646350777554489400361} a^{17} + \frac{199966544104265711417}{646350777554489400361} a^{16} + \frac{128091161842465794364}{646350777554489400361} a^{15} - \frac{245179678791811000472}{646350777554489400361} a^{14} + \frac{193385480230856006765}{646350777554489400361} a^{13} - \frac{309574705858308599210}{646350777554489400361} a^{12} + \frac{237716352430168967193}{646350777554489400361} a^{11} - \frac{233238504591547294785}{646350777554489400361} a^{10} - \frac{246669608163159715433}{646350777554489400361} a^{9} - \frac{110385583595575127059}{646350777554489400361} a^{8} + \frac{236927770451397644640}{646350777554489400361} a^{7} + \frac{194214051326046009977}{646350777554489400361} a^{6} + \frac{296445564726747282590}{646350777554489400361} a^{5} - \frac{127936345240679821106}{646350777554489400361} a^{4} - \frac{60315211270137514639}{646350777554489400361} a^{3} - \frac{160691481683804099624}{646350777554489400361} a^{2} + \frac{188720169883838503824}{646350777554489400361} a + \frac{223439666159188469853}{646350777554489400361}$

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:  $13$
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:  \( 192042.173919 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_5$ (as 20T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 8 conjugacy class representatives for $C_2^4:C_5$
Character table for $C_2^4:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.219503494144.1 x2, 10.6.219503494144.2

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

Sibling fields

Degree 10 siblings: data not computed
Degree 16 sibling: data not computed
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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$