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Properties

Label 20.20.1847542066...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{16}\cdot 5^{22}\cdot 7^{8}\cdot 29^{5}$
Root discriminant $51.68$
Ramified primes $2, 5, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2:F_5$ (as 20T22)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-41, 460, 100, -6755, 1655, 31774, -3400, -63410, 4050, 65020, -5160, -36760, 4245, 11500, -1700, -1933, 330, 160, -30, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 30*x^18 + 160*x^17 + 330*x^16 - 1933*x^15 - 1700*x^14 + 11500*x^13 + 4245*x^12 - 36760*x^11 - 5160*x^10 + 65020*x^9 + 4050*x^8 - 63410*x^7 - 3400*x^6 + 31774*x^5 + 1655*x^4 - 6755*x^3 + 100*x^2 + 460*x - 41)
 
gp: K = bnfinit(x^20 - 5*x^19 - 30*x^18 + 160*x^17 + 330*x^16 - 1933*x^15 - 1700*x^14 + 11500*x^13 + 4245*x^12 - 36760*x^11 - 5160*x^10 + 65020*x^9 + 4050*x^8 - 63410*x^7 - 3400*x^6 + 31774*x^5 + 1655*x^4 - 6755*x^3 + 100*x^2 + 460*x - 41, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 30 x^{18} + 160 x^{17} + 330 x^{16} - 1933 x^{15} - 1700 x^{14} + 11500 x^{13} + 4245 x^{12} - 36760 x^{11} - 5160 x^{10} + 65020 x^{9} + 4050 x^{8} - 63410 x^{7} - 3400 x^{6} + 31774 x^{5} + 1655 x^{4} - 6755 x^{3} + 100 x^{2} + 460 x - 41 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(18475420666617031250000000000000000=2^{16}\cdot 5^{22}\cdot 7^{8}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.68$
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:  $2, 5, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{287} a^{18} - \frac{2}{41} a^{17} + \frac{3}{41} a^{16} - \frac{127}{287} a^{15} - \frac{102}{287} a^{14} - \frac{100}{287} a^{13} - \frac{38}{287} a^{12} + \frac{113}{287} a^{11} + \frac{51}{287} a^{10} - \frac{61}{287} a^{9} - \frac{113}{287} a^{8} + \frac{10}{287} a^{7} + \frac{94}{287} a^{6} + \frac{143}{287} a^{5} + \frac{30}{287} a^{4} + \frac{115}{287} a^{3} + \frac{92}{287} a^{2} - \frac{136}{287} a - \frac{3}{7}$, $\frac{1}{2440723403310341801008721} a^{19} + \frac{4061909452732819685146}{2440723403310341801008721} a^{18} - \frac{160441232282159545349600}{348674771901477400144103} a^{17} + \frac{127101791071723988921827}{2440723403310341801008721} a^{16} + \frac{387335816449595009257948}{2440723403310341801008721} a^{15} + \frac{1178132599253240958810725}{2440723403310341801008721} a^{14} - \frac{7913501962541139814460}{26821136300113646164931} a^{13} - \frac{430346393986716754724182}{2440723403310341801008721} a^{12} - \frac{612296056879488576881555}{2440723403310341801008721} a^{11} - \frac{481104915125495118262211}{2440723403310341801008721} a^{10} - \frac{1055733429983414816789978}{2440723403310341801008721} a^{9} - \frac{99222572375748982326333}{2440723403310341801008721} a^{8} + \frac{556375297611779277270018}{2440723403310341801008721} a^{7} - \frac{27144551467450463591717}{78733013010011025838991} a^{6} - \frac{1050361570751418279461799}{2440723403310341801008721} a^{5} - \frac{9855620623184206197012}{348674771901477400144103} a^{4} - \frac{142745557887222856406462}{348674771901477400144103} a^{3} + \frac{594967106912697277524366}{2440723403310341801008721} a^{2} + \frac{37307441383582643279446}{78733013010011025838991} a + \frac{5916958838418507571308}{59529839105130287829481}$

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:  $19$
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:  \( 49401883665.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:F_5$ (as 20T22):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 14 conjugacy class representatives for $C_2^2:F_5$
Character table for $C_2^2:F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 5.5.2450000.1, 10.10.30012500000000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
5Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$