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Properties

Label 20.20.2947807911...1376.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{40}\cdot 401^{9}$
Root discriminant $59.36$
Ramified primes $2, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5:D_4$ (as 20T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![193, 2848, 14322, 20760, -45525, -142076, 5488, 264656, 100886, -207804, -111270, 80696, 50037, -15868, -11068, 1424, 1207, -40, -58, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 58*x^18 - 40*x^17 + 1207*x^16 + 1424*x^15 - 11068*x^14 - 15868*x^13 + 50037*x^12 + 80696*x^11 - 111270*x^10 - 207804*x^9 + 100886*x^8 + 264656*x^7 + 5488*x^6 - 142076*x^5 - 45525*x^4 + 20760*x^3 + 14322*x^2 + 2848*x + 193)
 
gp: K = bnfinit(x^20 - 58*x^18 - 40*x^17 + 1207*x^16 + 1424*x^15 - 11068*x^14 - 15868*x^13 + 50037*x^12 + 80696*x^11 - 111270*x^10 - 207804*x^9 + 100886*x^8 + 264656*x^7 + 5488*x^6 - 142076*x^5 - 45525*x^4 + 20760*x^3 + 14322*x^2 + 2848*x + 193, 1)
 

Normalized defining polynomial

\( x^{20} - 58 x^{18} - 40 x^{17} + 1207 x^{16} + 1424 x^{15} - 11068 x^{14} - 15868 x^{13} + 50037 x^{12} + 80696 x^{11} - 111270 x^{10} - 207804 x^{9} + 100886 x^{8} + 264656 x^{7} + 5488 x^{6} - 142076 x^{5} - 45525 x^{4} + 20760 x^{3} + 14322 x^{2} + 2848 x + 193 \)

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:  \(294780791174314477143981746737381376=2^{40}\cdot 401^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.36$
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:  $2, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{51} a^{18} - \frac{1}{17} a^{17} + \frac{7}{51} a^{16} - \frac{25}{51} a^{15} - \frac{3}{17} a^{14} + \frac{5}{51} a^{12} - \frac{22}{51} a^{11} - \frac{5}{51} a^{10} + \frac{7}{17} a^{9} - \frac{25}{51} a^{8} - \frac{1}{17} a^{7} - \frac{2}{17} a^{6} + \frac{20}{51} a^{5} - \frac{8}{51} a^{4} - \frac{19}{51} a^{3} - \frac{16}{51} a^{2} + \frac{7}{51} a - \frac{8}{51}$, $\frac{1}{68314986019747064309660380017363} a^{19} + \frac{494321827975360074373534721584}{68314986019747064309660380017363} a^{18} - \frac{19867956802244646264303071837192}{68314986019747064309660380017363} a^{17} + \frac{3326798637490786289610198702496}{22771662006582354769886793339121} a^{16} - \frac{27501457160687436845190835442434}{68314986019747064309660380017363} a^{15} + \frac{6026525764204236201509313784957}{22771662006582354769886793339121} a^{14} - \frac{26780527833230974512541366829554}{68314986019747064309660380017363} a^{13} - \frac{19342668932868959072530736197910}{68314986019747064309660380017363} a^{12} + \frac{5614320094828798017841138194854}{22771662006582354769886793339121} a^{11} - \frac{19508936973523320210608532264566}{68314986019747064309660380017363} a^{10} - \frac{13164901698963344437147911569644}{68314986019747064309660380017363} a^{9} + \frac{14169821807253809408587848137153}{68314986019747064309660380017363} a^{8} - \frac{9158408195607207541264457540955}{22771662006582354769886793339121} a^{7} + \frac{17671884092838338162857291899842}{68314986019747064309660380017363} a^{6} - \frac{4957843067271113432487294940063}{22771662006582354769886793339121} a^{5} + \frac{6497829807120413323526560426267}{22771662006582354769886793339121} a^{4} - \frac{8086058394246201839400708439838}{68314986019747064309660380017363} a^{3} + \frac{209280106377839488962604861548}{22771662006582354769886793339121} a^{2} + \frac{29087609035615845380029104111344}{68314986019747064309660380017363} a + \frac{25812869598131733372617218415836}{68314986019747064309660380017363}$

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:  \( 194057079592 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:D_4$ (as 20T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 13 conjugacy class representatives for $C_5:D_4$
Character table for $C_5:D_4$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.102656.1, 5.5.160801.1, 10.10.847280917741568.1

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

Sibling fields

Galois closure: data not computed
Degree 20 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
401Data not computed