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Properties

Label 20.0.76169967501...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 3^{19}\cdot 5^{12}$
Root discriminant $19.68$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $D_4\times F_5$ (as 20T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 0, -24, 0, 45, 0, 12, 0, 6, 0, -12, 0, -30, 0, -12, 0, 39, 0, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 12*x^18 + 39*x^16 - 12*x^14 - 30*x^12 - 12*x^10 + 6*x^8 + 12*x^6 + 45*x^4 - 24*x^2 + 3)
 
gp: K = bnfinit(x^20 - 12*x^18 + 39*x^16 - 12*x^14 - 30*x^12 - 12*x^10 + 6*x^8 + 12*x^6 + 45*x^4 - 24*x^2 + 3, 1)
 

Normalized defining polynomial

\( x^{20} - 12 x^{18} + 39 x^{16} - 12 x^{14} - 30 x^{12} - 12 x^{10} + 6 x^{8} + 12 x^{6} + 45 x^{4} - 24 x^{2} + 3 \)

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:  \(76169967501312000000000000=2^{28}\cdot 3^{19}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.68$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{5} - \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{3}{8} a - \frac{1}{8}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} + \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{3}{8} a^{4} + \frac{1}{8} a^{2} + \frac{3}{8}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{56} a^{16} + \frac{3}{56} a^{14} - \frac{3}{56} a^{12} - \frac{3}{56} a^{10} - \frac{3}{56} a^{8} + \frac{1}{56} a^{6} - \frac{5}{56} a^{4} - \frac{17}{56} a^{2} - \frac{3}{28}$, $\frac{1}{112} a^{17} - \frac{1}{112} a^{16} - \frac{1}{28} a^{15} + \frac{1}{28} a^{14} + \frac{1}{28} a^{13} + \frac{5}{56} a^{12} + \frac{1}{28} a^{11} - \frac{1}{28} a^{10} - \frac{5}{56} a^{9} + \frac{3}{14} a^{8} + \frac{1}{14} a^{7} - \frac{1}{14} a^{6} + \frac{1}{7} a^{5} + \frac{13}{56} a^{4} - \frac{3}{14} a^{3} - \frac{2}{7} a^{2} - \frac{27}{112} a + \frac{13}{112}$, $\frac{1}{112} a^{18} - \frac{1}{112} a^{16} + \frac{3}{56} a^{14} + \frac{1}{56} a^{12} - \frac{3}{28} a^{10} - \frac{1}{14} a^{8} + \frac{13}{56} a^{6} - \frac{9}{56} a^{4} + \frac{27}{112} a^{2} - \frac{39}{112}$, $\frac{1}{112} a^{19} - \frac{1}{112} a^{16} + \frac{1}{56} a^{15} + \frac{1}{28} a^{14} + \frac{3}{56} a^{13} + \frac{5}{56} a^{12} - \frac{1}{14} a^{11} - \frac{1}{28} a^{10} + \frac{5}{56} a^{9} - \frac{1}{28} a^{8} - \frac{11}{56} a^{7} - \frac{1}{14} a^{6} - \frac{1}{56} a^{5} + \frac{13}{56} a^{4} - \frac{53}{112} a^{3} - \frac{2}{7} a^{2} + \frac{9}{56} a + \frac{41}{112}$

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:  \( -\frac{45}{16} a^{18} + \frac{529}{16} a^{16} - \frac{813}{8} a^{14} + \frac{73}{8} a^{12} + 86 a^{10} + \frac{219}{4} a^{8} - \frac{27}{8} a^{6} - \frac{273}{8} a^{4} - \frac{2163}{16} a^{2} + \frac{563}{16} \) (order $6$)
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:  \( 251749.70025 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times F_5$ (as 20T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 25 conjugacy class representatives for $D_4\times F_5$
Character table for $D_4\times F_5$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.432.1, 5.1.162000.1, 10.0.78732000000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
3Data not computed
5Data not computed