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Properties

Label 20.10.7341802198...6875.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,5^{10}\cdot 13^{12}\cdot 19^{9}$
Root discriminant $39.20$
Ramified primes $5, 13, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![-1, -9, -7, 269, 1642, 31, -6997, -5488, 7388, 8861, -3758, -5838, 1627, 1786, -556, -268, 113, 25, -14, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 14*x^18 + 25*x^17 + 113*x^16 - 268*x^15 - 556*x^14 + 1786*x^13 + 1627*x^12 - 5838*x^11 - 3758*x^10 + 8861*x^9 + 7388*x^8 - 5488*x^7 - 6997*x^6 + 31*x^5 + 1642*x^4 + 269*x^3 - 7*x^2 - 9*x - 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 14*x^18 + 25*x^17 + 113*x^16 - 268*x^15 - 556*x^14 + 1786*x^13 + 1627*x^12 - 5838*x^11 - 3758*x^10 + 8861*x^9 + 7388*x^8 - 5488*x^7 - 6997*x^6 + 31*x^5 + 1642*x^4 + 269*x^3 - 7*x^2 - 9*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 14 x^{18} + 25 x^{17} + 113 x^{16} - 268 x^{15} - 556 x^{14} + 1786 x^{13} + 1627 x^{12} - 5838 x^{11} - 3758 x^{10} + 8861 x^{9} + 7388 x^{8} - 5488 x^{7} - 6997 x^{6} + 31 x^{5} + 1642 x^{4} + 269 x^{3} - 7 x^{2} - 9 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-73418021980786768815133779296875=-\,5^{10}\cdot 13^{12}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.20$
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:  $5, 13, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{58} a^{17} - \frac{7}{58} a^{16} - \frac{3}{29} a^{15} - \frac{6}{29} a^{14} + \frac{5}{29} a^{13} + \frac{5}{58} a^{12} - \frac{6}{29} a^{11} - \frac{4}{29} a^{10} + \frac{15}{58} a^{9} + \frac{7}{29} a^{8} - \frac{5}{29} a^{7} - \frac{5}{58} a^{6} + \frac{5}{29} a^{5} + \frac{14}{29} a^{4} + \frac{21}{58} a^{3} + \frac{3}{58} a^{2} + \frac{11}{58} a - \frac{23}{58}$, $\frac{1}{58} a^{18} + \frac{3}{58} a^{16} + \frac{2}{29} a^{15} + \frac{13}{58} a^{14} - \frac{6}{29} a^{13} - \frac{3}{29} a^{12} - \frac{5}{58} a^{11} - \frac{6}{29} a^{10} - \frac{13}{29} a^{9} + \frac{1}{58} a^{8} + \frac{6}{29} a^{7} + \frac{2}{29} a^{6} + \frac{11}{58} a^{5} + \frac{7}{29} a^{4} + \frac{5}{58} a^{3} + \frac{3}{58} a^{2} + \frac{25}{58} a - \frac{8}{29}$, $\frac{1}{134455124611407146327251934} a^{19} - \frac{35423296822679934102320}{67227562305703573163625967} a^{18} + \frac{37574051009799987098194}{67227562305703573163625967} a^{17} - \frac{17695949910536591080894719}{134455124611407146327251934} a^{16} - \frac{2470235885486827877525867}{67227562305703573163625967} a^{15} + \frac{18790873069613716415494665}{134455124611407146327251934} a^{14} - \frac{6399028156678914782056407}{134455124611407146327251934} a^{13} - \frac{19427386731527739312955769}{134455124611407146327251934} a^{12} + \frac{48799015143968119480195065}{134455124611407146327251934} a^{11} + \frac{27014753380626861860230069}{134455124611407146327251934} a^{10} - \frac{14311805852737749689679945}{134455124611407146327251934} a^{9} + \frac{11973792987292065790764201}{134455124611407146327251934} a^{8} + \frac{49190257283394976319864267}{134455124611407146327251934} a^{7} + \frac{19449565114854842771736091}{134455124611407146327251934} a^{6} - \frac{5841942517052113619956377}{134455124611407146327251934} a^{5} + \frac{11932755932856532540459851}{67227562305703573163625967} a^{4} + \frac{23549270043290585024559089}{134455124611407146327251934} a^{3} - \frac{18877279861034288783828775}{134455124611407146327251934} a^{2} + \frac{10402952687371703197157557}{67227562305703573163625967} a - \frac{16605400160041678186046429}{134455124611407146327251934}$

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:  $14$
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:  \( 670449501.541 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), 4.2.475.1, 5.5.19827925.1, 10.10.1965733049028125.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13Data not computed
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$