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Properties

Label 20.20.1802032470...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{16}\cdot 3^{10}\cdot 5^{31}$
Root discriminant $36.54$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, -35, 215, 120, -2121, 1735, 7145, -11480, -5515, 19881, -5515, -11480, 7145, 1735, -2121, 120, 215, -35, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 35*x^18 + 215*x^17 + 120*x^16 - 2121*x^15 + 1735*x^14 + 7145*x^13 - 11480*x^12 - 5515*x^11 + 19881*x^10 - 5515*x^9 - 11480*x^8 + 7145*x^7 + 1735*x^6 - 2121*x^5 + 120*x^4 + 215*x^3 - 35*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 - 35*x^18 + 215*x^17 + 120*x^16 - 2121*x^15 + 1735*x^14 + 7145*x^13 - 11480*x^12 - 5515*x^11 + 19881*x^10 - 5515*x^9 - 11480*x^8 + 7145*x^7 + 1735*x^6 - 2121*x^5 + 120*x^4 + 215*x^3 - 35*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 35 x^{18} + 215 x^{17} + 120 x^{16} - 2121 x^{15} + 1735 x^{14} + 7145 x^{13} - 11480 x^{12} - 5515 x^{11} + 19881 x^{10} - 5515 x^{9} - 11480 x^{8} + 7145 x^{7} + 1735 x^{6} - 2121 x^{5} + 120 x^{4} + 215 x^{3} - 35 x^{2} - 5 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(18020324707031250000000000000000=2^{16}\cdot 3^{10}\cdot 5^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.54$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{3} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{5} + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{3} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{75} a^{14} - \frac{1}{75} a^{13} + \frac{2}{25} a^{12} + \frac{4}{75} a^{11} - \frac{4}{75} a^{10} - \frac{1}{25} a^{9} - \frac{2}{75} a^{8} + \frac{2}{75} a^{7} - \frac{32}{75} a^{6} - \frac{11}{25} a^{5} - \frac{4}{75} a^{4} - \frac{26}{75} a^{3} + \frac{7}{25} a^{2} + \frac{14}{75} a - \frac{29}{75}$, $\frac{1}{75} a^{15} + \frac{1}{15} a^{13} - \frac{1}{15} a^{12} - \frac{7}{75} a^{10} - \frac{1}{15} a^{9} - \frac{4}{15} a^{6} + \frac{23}{75} a^{5} - \frac{2}{5} a^{4} - \frac{1}{15} a^{3} + \frac{1}{15} a^{2} + \frac{2}{5} a + \frac{31}{75}$, $\frac{1}{75} a^{16} + \frac{1}{25} a^{11} - \frac{1}{15} a^{8} + \frac{6}{25} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{3} - \frac{3}{25} a + \frac{2}{15}$, $\frac{1}{75} a^{17} + \frac{1}{25} a^{12} - \frac{1}{15} a^{9} + \frac{1}{25} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{8}{25} a^{2} - \frac{4}{15} a - \frac{1}{5}$, $\frac{1}{74325} a^{18} + \frac{74}{24775} a^{17} - \frac{61}{24775} a^{16} + \frac{74}{74325} a^{15} + \frac{254}{74325} a^{14} + \frac{1949}{74325} a^{13} - \frac{1202}{14865} a^{12} - \frac{5368}{74325} a^{11} + \frac{2272}{24775} a^{10} - \frac{5812}{74325} a^{9} + \frac{233}{2973} a^{8} - \frac{3386}{74325} a^{7} + \frac{1918}{74325} a^{6} + \frac{5543}{14865} a^{5} + \frac{5209}{74325} a^{4} + \frac{8002}{74325} a^{3} - \frac{30904}{74325} a^{2} + \frac{16078}{74325} a - \frac{32702}{74325}$, $\frac{1}{74325} a^{19} + \frac{83}{74325} a^{17} + \frac{23}{24775} a^{16} - \frac{106}{24775} a^{15} + \frac{22}{24775} a^{14} + \frac{769}{24775} a^{13} + \frac{304}{24775} a^{12} + \frac{1384}{74325} a^{11} - \frac{1572}{24775} a^{10} + \frac{2834}{74325} a^{9} + \frac{558}{24775} a^{8} + \frac{6}{991} a^{7} - \frac{8158}{24775} a^{6} + \frac{4839}{24775} a^{5} + \frac{10298}{24775} a^{4} - \frac{33458}{74325} a^{3} - \frac{10168}{24775} a^{2} - \frac{15589}{74325} a - \frac{1204}{4955}$

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:  \( 2193798277.46 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 5.5.11250000.1 x5, 10.10.632812500000000.1 x5

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

Sibling fields

Degree 5 sibling: 5.5.11250000.1
Degree 10 sibling: 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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5Data not computed