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Properties

Label 20.2.35136158630...1875.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,5^{26}\cdot 11^{9}$
Root discriminant $23.84$
Ramified primes $5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{20}$ (as 20T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -45, -170, -180, 90, 342, 550, 450, -260, -630, -312, -225, 205, 360, 70, -18, 15, 0, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 + 15*x^16 - 18*x^15 + 70*x^14 + 360*x^13 + 205*x^12 - 225*x^11 - 312*x^10 - 630*x^9 - 260*x^8 + 450*x^7 + 550*x^6 + 342*x^5 + 90*x^4 - 180*x^3 - 170*x^2 - 45*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^18 + 15*x^16 - 18*x^15 + 70*x^14 + 360*x^13 + 205*x^12 - 225*x^11 - 312*x^10 - 630*x^9 - 260*x^8 + 450*x^7 + 550*x^6 + 342*x^5 + 90*x^4 - 180*x^3 - 170*x^2 - 45*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{18} + 15 x^{16} - 18 x^{15} + 70 x^{14} + 360 x^{13} + 205 x^{12} - 225 x^{11} - 312 x^{10} - 630 x^{9} - 260 x^{8} + 450 x^{7} + 550 x^{6} + 342 x^{5} + 90 x^{4} - 180 x^{3} - 170 x^{2} - 45 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-3513615863025188446044921875=-\,5^{26}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.84$
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:  $5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{9} a^{6} + \frac{2}{9} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} + \frac{1}{9} a^{7} + \frac{2}{9} a^{3} + \frac{1}{9} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{1}{9} a^{6} + \frac{2}{9} a^{4} + \frac{1}{3} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{11} + \frac{1}{9} a^{9} + \frac{1}{9} a^{7} + \frac{2}{9} a^{5} + \frac{1}{3} a^{3} + \frac{1}{9} a$, $\frac{1}{27} a^{16} - \frac{1}{27} a^{15} - \frac{1}{27} a^{14} - \frac{1}{27} a^{12} + \frac{4}{27} a^{11} + \frac{2}{27} a^{10} - \frac{1}{27} a^{9} - \frac{1}{9} a^{8} + \frac{2}{27} a^{7} + \frac{4}{27} a^{6} + \frac{1}{27} a^{5} + \frac{4}{27} a^{4} - \frac{1}{9} a^{3} + \frac{1}{27} a^{2} + \frac{2}{27} a + \frac{5}{27}$, $\frac{1}{27} a^{17} + \frac{1}{27} a^{15} - \frac{1}{27} a^{14} - \frac{1}{27} a^{13} + \frac{1}{9} a^{11} + \frac{4}{27} a^{10} - \frac{1}{27} a^{9} - \frac{1}{27} a^{8} + \frac{2}{27} a^{6} + \frac{2}{27} a^{5} + \frac{1}{27} a^{4} - \frac{2}{27} a^{3} - \frac{1}{9} a^{2} + \frac{1}{27} a + \frac{2}{27}$, $\frac{1}{2997} a^{18} - \frac{2}{999} a^{17} + \frac{11}{999} a^{16} + \frac{41}{999} a^{15} - \frac{34}{999} a^{14} - \frac{16}{999} a^{13} - \frac{68}{2997} a^{12} + \frac{113}{999} a^{11} - \frac{7}{111} a^{10} + \frac{53}{333} a^{9} - \frac{1}{27} a^{8} + \frac{145}{999} a^{7} - \frac{428}{2997} a^{6} + \frac{445}{999} a^{5} + \frac{188}{999} a^{4} - \frac{161}{999} a^{3} - \frac{110}{333} a^{2} - \frac{178}{999} a + \frac{277}{2997}$, $\frac{1}{67236170400646479} a^{19} + \frac{56138052937}{1817193794612067} a^{18} + \frac{388978308735994}{22412056800215493} a^{17} + \frac{365159604173804}{22412056800215493} a^{16} + \frac{915606227688292}{22412056800215493} a^{15} - \frac{923985360380014}{22412056800215493} a^{14} - \frac{2313448715998100}{67236170400646479} a^{13} - \frac{649304977128860}{67236170400646479} a^{12} + \frac{162112103457221}{2490228533357277} a^{11} - \frac{907369606338406}{7470685600071831} a^{10} + \frac{313798297020851}{7470685600071831} a^{9} + \frac{2912225828132192}{22412056800215493} a^{8} - \frac{8162028514514588}{67236170400646479} a^{7} + \frac{3239030599956529}{67236170400646479} a^{6} - \frac{1368950132822906}{7470685600071831} a^{5} - \frac{1762033761835267}{22412056800215493} a^{4} + \frac{3696321298827851}{22412056800215493} a^{3} - \frac{81473006163802}{393193978951149} a^{2} - \frac{8991664892197619}{67236170400646479} a + \frac{404392804294915}{67236170400646479}$

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:  $10$
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:  \( 1089848.04881 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{20}$ (as 20T10):

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 $D_{20}$
Character table for $D_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 5.1.1890625.1, 10.2.17872314453125.1

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 $20$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ R $20$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
5Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$