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Properties

Label 20.0.87161254882...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{8}\cdot 5^{23}\cdot 13^{4}$
Root discriminant $14.03$
Ramified primes $2, 5, 13$
Class number $1$
Class group Trivial
Galois group $C_4\times S_5$ (as 20T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, -5, 15, -11, -10, 25, -15, -10, 21, -10, -15, 25, -10, -11, 15, -5, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^17 + 15*x^16 - 11*x^15 - 10*x^14 + 25*x^13 - 15*x^12 - 10*x^11 + 21*x^10 - 10*x^9 - 15*x^8 + 25*x^7 - 10*x^6 - 11*x^5 + 15*x^4 - 5*x^3 + 1)
 
gp: K = bnfinit(x^20 - 5*x^17 + 15*x^16 - 11*x^15 - 10*x^14 + 25*x^13 - 15*x^12 - 10*x^11 + 21*x^10 - 10*x^9 - 15*x^8 + 25*x^7 - 10*x^6 - 11*x^5 + 15*x^4 - 5*x^3 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{17} + 15 x^{16} - 11 x^{15} - 10 x^{14} + 25 x^{13} - 15 x^{12} - 10 x^{11} + 21 x^{10} - 10 x^{9} - 15 x^{8} + 25 x^{7} - 10 x^{6} - 11 x^{5} + 15 x^{4} - 5 x^{3} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(87161254882812500000000=2^{8}\cdot 5^{23}\cdot 13^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.03$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{101} a^{18} + \frac{11}{101} a^{17} + \frac{19}{101} a^{16} - \frac{9}{101} a^{15} - \frac{2}{101} a^{14} - \frac{24}{101} a^{13} + \frac{31}{101} a^{12} - \frac{14}{101} a^{11} + \frac{2}{101} a^{10} + \frac{26}{101} a^{9} + \frac{2}{101} a^{8} - \frac{14}{101} a^{7} + \frac{31}{101} a^{6} - \frac{24}{101} a^{5} - \frac{2}{101} a^{4} - \frac{9}{101} a^{3} + \frac{19}{101} a^{2} + \frac{11}{101} a + \frac{1}{101}$, $\frac{1}{101} a^{19} - \frac{1}{101} a^{17} - \frac{16}{101} a^{16} - \frac{4}{101} a^{15} - \frac{2}{101} a^{14} - \frac{8}{101} a^{13} + \frac{49}{101} a^{12} - \frac{46}{101} a^{11} + \frac{4}{101} a^{10} + \frac{19}{101} a^{9} - \frac{36}{101} a^{8} - \frac{17}{101} a^{7} + \frac{39}{101} a^{6} - \frac{41}{101} a^{5} + \frac{13}{101} a^{4} + \frac{17}{101} a^{3} + \frac{4}{101} a^{2} - \frac{19}{101} a - \frac{11}{101}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1632}{101} a^{19} - \frac{1961}{101} a^{18} - \frac{1052}{101} a^{17} + \frac{8750}{101} a^{16} - \frac{12789}{101} a^{15} - \frac{4227}{101} a^{14} + \frac{19922}{101} a^{13} - \frac{11176}{101} a^{12} - \frac{4231}{101} a^{11} + \frac{16719}{101} a^{10} - \frac{8870}{101} a^{9} - \frac{6881}{101} a^{8} + \frac{20252}{101} a^{7} - \frac{9400}{101} a^{6} - \frac{9951}{101} a^{5} + \frac{9875}{101} a^{4} - \frac{2924}{101} a^{3} - \frac{1569}{101} a^{2} - \frac{1269}{101} a + \frac{235}{101} \) (order $10$)
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:  \( 6920.92367284 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times S_5$ (as 20T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 480
The 28 conjugacy class representatives for $C_4\times S_5$
Character table for $C_4\times S_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.3.162500.1, 10.6.132031250000.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 24 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 $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.12.0.1$x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$$1$$12$$0$$C_{12}$$[\ ]^{12}$
5Data not computed
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$