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Properties

Label 20.4.76572488767...8161.1
Degree $20$
Signature $[4, 8]$
Discriminant $13^{4}\cdot 401^{9}$
Root discriminant $24.79$
Ramified primes $13, 401$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^4:D_5$ (as 20T87)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-31, -176, -565, -1041, -1281, -726, -290, -643, -207, 609, 286, -106, 88, 39, 10, 35, -14, -7, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - x^18 - 7*x^17 - 14*x^16 + 35*x^15 + 10*x^14 + 39*x^13 + 88*x^12 - 106*x^11 + 286*x^10 + 609*x^9 - 207*x^8 - 643*x^7 - 290*x^6 - 726*x^5 - 1281*x^4 - 1041*x^3 - 565*x^2 - 176*x - 31)
 
gp: K = bnfinit(x^20 - x^19 - x^18 - 7*x^17 - 14*x^16 + 35*x^15 + 10*x^14 + 39*x^13 + 88*x^12 - 106*x^11 + 286*x^10 + 609*x^9 - 207*x^8 - 643*x^7 - 290*x^6 - 726*x^5 - 1281*x^4 - 1041*x^3 - 565*x^2 - 176*x - 31, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - x^{18} - 7 x^{17} - 14 x^{16} + 35 x^{15} + 10 x^{14} + 39 x^{13} + 88 x^{12} - 106 x^{11} + 286 x^{10} + 609 x^{9} - 207 x^{8} - 643 x^{7} - 290 x^{6} - 726 x^{5} - 1281 x^{4} - 1041 x^{3} - 565 x^{2} - 176 x - 31 \)

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:  \(7657248876721128710150208161=13^{4}\cdot 401^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.79$
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:  $13, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{4}{13} a^{13} - \frac{1}{13} a^{12} + \frac{3}{13} a^{10} - \frac{2}{13} a^{9} + \frac{1}{13} a^{8} + \frac{5}{13} a^{7} + \frac{2}{13} a^{6} - \frac{3}{13} a^{5} - \frac{2}{13} a^{4} - \frac{1}{13} a^{3} - \frac{2}{13} a^{2} + \frac{5}{13} a + \frac{6}{13}$, $\frac{1}{13} a^{15} - \frac{4}{13} a^{13} + \frac{4}{13} a^{12} + \frac{3}{13} a^{11} - \frac{1}{13} a^{10} - \frac{4}{13} a^{9} + \frac{1}{13} a^{8} - \frac{5}{13} a^{7} + \frac{2}{13} a^{6} - \frac{3}{13} a^{5} - \frac{6}{13} a^{4} + \frac{2}{13} a^{3} - \frac{1}{13} a + \frac{2}{13}$, $\frac{1}{13} a^{16} - \frac{6}{13} a^{13} - \frac{1}{13} a^{12} - \frac{1}{13} a^{11} - \frac{5}{13} a^{10} + \frac{6}{13} a^{9} - \frac{1}{13} a^{8} - \frac{4}{13} a^{7} + \frac{5}{13} a^{6} - \frac{5}{13} a^{5} - \frac{6}{13} a^{4} - \frac{4}{13} a^{3} + \frac{4}{13} a^{2} - \frac{4}{13} a - \frac{2}{13}$, $\frac{1}{13} a^{17} - \frac{3}{13} a^{13} + \frac{6}{13} a^{12} - \frac{5}{13} a^{11} - \frac{2}{13} a^{10} + \frac{2}{13} a^{8} - \frac{4}{13} a^{7} - \frac{6}{13} a^{6} + \frac{2}{13} a^{5} - \frac{3}{13} a^{4} - \frac{2}{13} a^{3} - \frac{3}{13} a^{2} + \frac{2}{13} a - \frac{3}{13}$, $\frac{1}{13} a^{18} + \frac{5}{13} a^{13} + \frac{5}{13} a^{12} - \frac{2}{13} a^{11} - \frac{4}{13} a^{10} - \frac{4}{13} a^{9} - \frac{1}{13} a^{8} - \frac{4}{13} a^{7} - \frac{5}{13} a^{6} + \frac{1}{13} a^{5} + \frac{5}{13} a^{4} - \frac{6}{13} a^{3} - \frac{4}{13} a^{2} - \frac{1}{13} a + \frac{5}{13}$, $\frac{1}{11684283192758837194172717} a^{19} + \frac{7754786962468412835296}{11684283192758837194172717} a^{18} - \frac{331132389476796734851801}{11684283192758837194172717} a^{17} + \frac{50912438803337887357677}{11684283192758837194172717} a^{16} + \frac{41350317899986109915253}{11684283192758837194172717} a^{15} - \frac{102821816712694338219327}{11684283192758837194172717} a^{14} - \frac{5730854269499993797976463}{11684283192758837194172717} a^{13} + \frac{4757387502896657780722003}{11684283192758837194172717} a^{12} + \frac{3414218092498245684483770}{11684283192758837194172717} a^{11} - \frac{3294666740454982096943014}{11684283192758837194172717} a^{10} - \frac{1609765766006295496757906}{11684283192758837194172717} a^{9} + \frac{116149546026703481455721}{315791437642130734977641} a^{8} - \frac{11114944011080217045820}{11684283192758837194172717} a^{7} + \frac{4260889118496089585305009}{11684283192758837194172717} a^{6} + \frac{1493382513167955880278119}{11684283192758837194172717} a^{5} + \frac{3456144124384889061473797}{11684283192758837194172717} a^{4} + \frac{4707016158927800675997436}{11684283192758837194172717} a^{3} - \frac{3867506304230999080724622}{11684283192758837194172717} a^{2} + \frac{2630061431789088038493117}{11684283192758837194172717} a + \frac{971591676425880584691155}{11684283192758837194172717}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1121549.32732 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T87):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

5.5.160801.1, 10.2.4369826510569.1

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed