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Properties

Label 20.12.1304707890...2704.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{10}\cdot 3^{10}\cdot 11^{18}\cdot 197^{2}$
Root discriminant $35.96$
Ramified primes $2, 3, 11, 197$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T262

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![397, -4705, 15804, -20406, -583, 34046, -36704, 4815, 9118, 5610, -9503, -1232, 6324, -3622, 157, 760, -308, 21, 19, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 19*x^18 + 21*x^17 - 308*x^16 + 760*x^15 + 157*x^14 - 3622*x^13 + 6324*x^12 - 1232*x^11 - 9503*x^10 + 5610*x^9 + 9118*x^8 + 4815*x^7 - 36704*x^6 + 34046*x^5 - 583*x^4 - 20406*x^3 + 15804*x^2 - 4705*x + 397)
 
gp: K = bnfinit(x^20 - 8*x^19 + 19*x^18 + 21*x^17 - 308*x^16 + 760*x^15 + 157*x^14 - 3622*x^13 + 6324*x^12 - 1232*x^11 - 9503*x^10 + 5610*x^9 + 9118*x^8 + 4815*x^7 - 36704*x^6 + 34046*x^5 - 583*x^4 - 20406*x^3 + 15804*x^2 - 4705*x + 397, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 19 x^{18} + 21 x^{17} - 308 x^{16} + 760 x^{15} + 157 x^{14} - 3622 x^{13} + 6324 x^{12} - 1232 x^{11} - 9503 x^{10} + 5610 x^{9} + 9118 x^{8} + 4815 x^{7} - 36704 x^{6} + 34046 x^{5} - 583 x^{4} - 20406 x^{3} + 15804 x^{2} - 4705 x + 397 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13047078908784463218470268232704=2^{10}\cdot 3^{10}\cdot 11^{18}\cdot 197^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.96$
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:  $2, 3, 11, 197$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{67} a^{18} - \frac{4}{67} a^{17} - \frac{1}{67} a^{16} + \frac{33}{67} a^{15} + \frac{29}{67} a^{14} + \frac{7}{67} a^{13} + \frac{2}{67} a^{12} - \frac{24}{67} a^{11} - \frac{11}{67} a^{10} + \frac{26}{67} a^{9} + \frac{25}{67} a^{8} - \frac{22}{67} a^{7} + \frac{19}{67} a^{6} + \frac{21}{67} a^{5} + \frac{20}{67} a^{4} + \frac{6}{67} a^{3} + \frac{31}{67} a^{2} - \frac{5}{67} a - \frac{18}{67}$, $\frac{1}{1044201176711848952889528834277412845729} a^{19} + \frac{3473219980725269303932740789199266438}{1044201176711848952889528834277412845729} a^{18} - \frac{468537098633898396349617250338252989396}{1044201176711848952889528834277412845729} a^{17} + \frac{321100766768549923458207921957992034158}{1044201176711848952889528834277412845729} a^{16} - \frac{400648069193215520813776286037627521531}{1044201176711848952889528834277412845729} a^{15} + \frac{196069872746268639015696803397597286205}{1044201176711848952889528834277412845729} a^{14} + \frac{182431684124421540792412673746346543904}{1044201176711848952889528834277412845729} a^{13} - \frac{366058886185206985415255828727149280984}{1044201176711848952889528834277412845729} a^{12} - \frac{215112686381542256139498328680989381724}{1044201176711848952889528834277412845729} a^{11} - \frac{426087556425045980042719350913574966563}{1044201176711848952889528834277412845729} a^{10} + \frac{170858636750446878613333771370196583330}{1044201176711848952889528834277412845729} a^{9} + \frac{513236785133014008840625483980502033847}{1044201176711848952889528834277412845729} a^{8} - \frac{473450819708437625376654777874719625962}{1044201176711848952889528834277412845729} a^{7} - \frac{409060070418321358324409033262418628122}{1044201176711848952889528834277412845729} a^{6} + \frac{162006007326429912162297209279638420948}{1044201176711848952889528834277412845729} a^{5} - \frac{220580325789939907187520551151819511524}{1044201176711848952889528834277412845729} a^{4} - \frac{364091283815710647196537899536839140655}{1044201176711848952889528834277412845729} a^{3} + \frac{6320845293210539172993317570882106368}{15585092189729088849097445287722579787} a^{2} + \frac{202576870071744957120297934928814564688}{1044201176711848952889528834277412845729} a - \frac{139161916019302680085023331776021663562}{1044201176711848952889528834277412845729}$

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:  $15$
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:  \( 215903278.765 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T262:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2560
The 40 conjugacy class representatives for t20n262
Character table for t20n262 is not computed

Intermediate fields

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

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

Sibling fields

Degree 20 siblings: data not computed
Degree 32 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 R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.2$x^{10} - 5 x^{8} + 10 x^{6} - 2 x^{4} - 11 x^{2} + 39$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$197$$\Q_{197}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{197}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{197}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{197}$$x + 2$$1$$1$$0$Trivial$[\ ]$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$