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Properties

Label 20.12.7053647688...0561.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{8}\cdot 401^{10}$
Root discriminant $31.08$
Ramified primes $3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4:D_5$ (as 20T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -72, -24, 504, -142, -1436, 1324, 1432, -2919, 676, 1178, -1120, 732, -40, -350, 220, -60, -8, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 20*x^18 - 8*x^17 - 60*x^16 + 220*x^15 - 350*x^14 - 40*x^13 + 732*x^12 - 1120*x^11 + 1178*x^10 + 676*x^9 - 2919*x^8 + 1432*x^7 + 1324*x^6 - 1436*x^5 - 142*x^4 + 504*x^3 - 24*x^2 - 72*x + 9)
 
gp: K = bnfinit(x^20 - 8*x^19 + 20*x^18 - 8*x^17 - 60*x^16 + 220*x^15 - 350*x^14 - 40*x^13 + 732*x^12 - 1120*x^11 + 1178*x^10 + 676*x^9 - 2919*x^8 + 1432*x^7 + 1324*x^6 - 1436*x^5 - 142*x^4 + 504*x^3 - 24*x^2 - 72*x + 9, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 20 x^{18} - 8 x^{17} - 60 x^{16} + 220 x^{15} - 350 x^{14} - 40 x^{13} + 732 x^{12} - 1120 x^{11} + 1178 x^{10} + 676 x^{9} - 2919 x^{8} + 1432 x^{7} + 1324 x^{6} - 1436 x^{5} - 142 x^{4} + 504 x^{3} - 24 x^{2} - 72 x + 9 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(705364768808763611651745450561=3^{8}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.08$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{12} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{5}{12} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{12} a^{11} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{12} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{4} a$, $\frac{1}{12} a^{12} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{12} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{12} a^{13} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{12} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{12} a^{14} - \frac{1}{4} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{12} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2}$, $\frac{1}{24} a^{15} - \frac{1}{24} a^{12} - \frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{6} a^{8} + \frac{1}{12} a^{7} + \frac{5}{24} a^{6} - \frac{5}{24} a^{5} - \frac{1}{24} a^{4} - \frac{1}{24} a^{2} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{24} a^{16} - \frac{1}{24} a^{13} - \frac{1}{24} a^{11} - \frac{1}{24} a^{10} - \frac{1}{6} a^{9} + \frac{1}{12} a^{8} + \frac{5}{24} a^{7} - \frac{5}{24} a^{6} - \frac{1}{24} a^{5} - \frac{1}{24} a^{3} - \frac{1}{2} a^{2} + \frac{3}{8} a$, $\frac{1}{24} a^{17} - \frac{1}{24} a^{14} - \frac{1}{24} a^{12} - \frac{1}{24} a^{11} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{1}{24} a^{7} - \frac{1}{24} a^{6} - \frac{1}{6} a^{5} - \frac{3}{8} a^{4} - \frac{1}{6} a^{3} + \frac{1}{24} a^{2}$, $\frac{1}{72} a^{18} - \frac{1}{72} a^{17} - \frac{1}{72} a^{16} - \frac{1}{72} a^{15} + \frac{1}{72} a^{14} - \frac{1}{36} a^{13} - \frac{1}{24} a^{10} - \frac{1}{72} a^{9} - \frac{1}{9} a^{8} + \frac{13}{72} a^{7} - \frac{5}{36} a^{6} + \frac{5}{36} a^{5} - \frac{29}{72} a^{4} - \frac{1}{3} a^{3} - \frac{1}{8} a^{2} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{167546728316952} a^{19} + \frac{75206856299}{27924454719492} a^{18} - \frac{334757305577}{167546728316952} a^{17} + \frac{2318257856587}{167546728316952} a^{16} - \frac{995392429289}{55848909438984} a^{15} + \frac{2191756856323}{83773364158476} a^{14} - \frac{3356622199343}{167546728316952} a^{13} - \frac{94957075490}{2327037893291} a^{12} + \frac{762906025941}{18616303146328} a^{11} - \frac{2810048789813}{83773364158476} a^{10} + \frac{3337424100673}{13962227359746} a^{9} - \frac{5108667973967}{83773364158476} a^{8} - \frac{6176035687997}{55848909438984} a^{7} - \frac{4091106020491}{55848909438984} a^{6} - \frac{37881584583415}{167546728316952} a^{5} - \frac{67482422821091}{167546728316952} a^{4} - \frac{1689859096921}{27924454719492} a^{3} + \frac{949559036871}{2327037893291} a^{2} + \frac{1254237308421}{9308151573164} a - \frac{4575371524693}{18616303146328}$

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:  $15$
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:  \( 77469369.7374 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:D_5$ (as 20T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 10 conjugacy class representatives for $C_2^4:D_5$
Character table for $C_2^4:D_5$

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.6.2094413889681.1, 10.6.839859969762081.2, 10.10.10368641602001.1

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 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{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}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed