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Properties

Label 20.4.62402607650...0625.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{10}\cdot 5^{10}\cdot 641^{5}$
Root discriminant $19.49$
Ramified primes $3, 5, 641$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times D_5\wr C_2$ (as 20T92)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 32, 171, 377, 625, 534, 363, -200, -349, -395, 43, 72, 286, 87, -278, 86, 23, -36, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 24*x^18 - 36*x^17 + 23*x^16 + 86*x^15 - 278*x^14 + 87*x^13 + 286*x^12 + 72*x^11 + 43*x^10 - 395*x^9 - 349*x^8 - 200*x^7 + 363*x^6 + 534*x^5 + 625*x^4 + 377*x^3 + 171*x^2 + 32*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 24*x^18 - 36*x^17 + 23*x^16 + 86*x^15 - 278*x^14 + 87*x^13 + 286*x^12 + 72*x^11 + 43*x^10 - 395*x^9 - 349*x^8 - 200*x^7 + 363*x^6 + 534*x^5 + 625*x^4 + 377*x^3 + 171*x^2 + 32*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 24 x^{18} - 36 x^{17} + 23 x^{16} + 86 x^{15} - 278 x^{14} + 87 x^{13} + 286 x^{12} + 72 x^{11} + 43 x^{10} - 395 x^{9} - 349 x^{8} - 200 x^{7} + 363 x^{6} + 534 x^{5} + 625 x^{4} + 377 x^{3} + 171 x^{2} + 32 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(62402607650205857900390625=3^{10}\cdot 5^{10}\cdot 641^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.49$
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:  $3, 5, 641$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{13} a^{17} - \frac{3}{13} a^{16} - \frac{5}{13} a^{15} - \frac{3}{13} a^{14} + \frac{4}{13} a^{13} + \frac{3}{13} a^{12} - \frac{3}{13} a^{11} + \frac{3}{13} a^{10} + \frac{1}{13} a^{9} - \frac{1}{13} a^{7} + \frac{6}{13} a^{6} - \frac{6}{13} a^{5} - \frac{5}{13} a^{4} - \frac{2}{13} a^{3} + \frac{4}{13} a^{2} - \frac{5}{13} a + \frac{4}{13}$, $\frac{1}{13} a^{18} - \frac{1}{13} a^{16} - \frac{5}{13} a^{15} - \frac{5}{13} a^{14} + \frac{2}{13} a^{13} + \frac{6}{13} a^{12} - \frac{6}{13} a^{11} - \frac{3}{13} a^{10} + \frac{3}{13} a^{9} - \frac{1}{13} a^{8} + \frac{3}{13} a^{7} - \frac{1}{13} a^{6} + \frac{3}{13} a^{5} - \frac{4}{13} a^{4} - \frac{2}{13} a^{3} - \frac{6}{13} a^{2} + \frac{2}{13} a - \frac{1}{13}$, $\frac{1}{13053949258745568264989729} a^{19} + \frac{78614445481161157170505}{13053949258745568264989729} a^{18} - \frac{195156046043531968305562}{13053949258745568264989729} a^{17} - \frac{5498541470318575813452547}{13053949258745568264989729} a^{16} - \frac{4714920086404153730841507}{13053949258745568264989729} a^{15} + \frac{4493247820600729681434906}{13053949258745568264989729} a^{14} - \frac{5974529459815559239461357}{13053949258745568264989729} a^{13} - \frac{5437494299599266345660400}{13053949258745568264989729} a^{12} + \frac{2360739530836888643452923}{13053949258745568264989729} a^{11} + \frac{2123925931148065435915974}{13053949258745568264989729} a^{10} - \frac{4760885937381452753438509}{13053949258745568264989729} a^{9} + \frac{6007041397162563930662357}{13053949258745568264989729} a^{8} - \frac{5203941549841346738810184}{13053949258745568264989729} a^{7} - \frac{4715314090419597959638041}{13053949258745568264989729} a^{6} - \frac{6508265907630874699202585}{13053949258745568264989729} a^{5} - \frac{2499666923307742410300944}{13053949258745568264989729} a^{4} + \frac{6028196131273063874865632}{13053949258745568264989729} a^{3} + \frac{4331475707201389739844567}{13053949258745568264989729} a^{2} - \frac{3853513996681294023800485}{13053949258745568264989729} a - \frac{5855499821856660420967714}{13053949258745568264989729}$

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:  $11$
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:  \( 83555.0948812 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_5\wr C_2$ (as 20T92):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $C_2\times D_5\wr C_2$
Character table for $C_2\times D_5\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.144225.1, 10.2.1284003125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
641Data not computed