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Properties

Label 20.0.69974789420...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 3319^{4}$
Root discriminant $19.60$
Ramified primes $3, 5, 3319$
Class number $1$
Class group Trivial
Galois group $C_2\times D_5\wr C_2$ (as 20T100)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, 128, -172, 139, -217, 304, -301, 230, -211, 234, -184, 116, -91, 76, -52, 28, -13, 8, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 8*x^18 - 13*x^17 + 28*x^16 - 52*x^15 + 76*x^14 - 91*x^13 + 116*x^12 - 184*x^11 + 234*x^10 - 211*x^9 + 230*x^8 - 301*x^7 + 304*x^6 - 217*x^5 + 139*x^4 - 172*x^3 + 128*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 + 8*x^18 - 13*x^17 + 28*x^16 - 52*x^15 + 76*x^14 - 91*x^13 + 116*x^12 - 184*x^11 + 234*x^10 - 211*x^9 + 230*x^8 - 301*x^7 + 304*x^6 - 217*x^5 + 139*x^4 - 172*x^3 + 128*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 8 x^{18} - 13 x^{17} + 28 x^{16} - 52 x^{15} + 76 x^{14} - 91 x^{13} + 116 x^{12} - 184 x^{11} + 234 x^{10} - 211 x^{9} + 230 x^{8} - 301 x^{7} + 304 x^{6} - 217 x^{5} + 139 x^{4} - 172 x^{3} + 128 x^{2} - 12 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(69974789420587753212890625=3^{10}\cdot 5^{10}\cdot 3319^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.60$
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:  $3, 5, 3319$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2678433} a^{18} - \frac{760777}{2678433} a^{17} + \frac{884606}{2678433} a^{16} + \frac{920551}{2678433} a^{15} + \frac{181}{807} a^{14} - \frac{181779}{892811} a^{13} - \frac{392268}{892811} a^{12} - \frac{284914}{2678433} a^{11} + \frac{33854}{892811} a^{10} - \frac{73524}{892811} a^{9} - \frac{146288}{892811} a^{8} + \frac{1161077}{2678433} a^{7} - \frac{334532}{892811} a^{6} - \frac{279360}{892811} a^{5} - \frac{754601}{2678433} a^{4} - \frac{61388}{2678433} a^{3} - \frac{129796}{2678433} a^{2} + \frac{940460}{2678433} a + \frac{201967}{2678433}$, $\frac{1}{18868657853079} a^{19} + \frac{1935430}{18868657853079} a^{18} - \frac{1159736635628}{6289552617693} a^{17} + \frac{1852237027406}{18868657853079} a^{16} - \frac{1377839528188}{6289552617693} a^{15} + \frac{3887280207869}{18868657853079} a^{14} + \frac{2198213819792}{6289552617693} a^{13} - \frac{7657059642952}{18868657853079} a^{12} - \frac{4541879485130}{18868657853079} a^{11} + \frac{2915119767391}{6289552617693} a^{10} + \frac{2975966843245}{6289552617693} a^{9} - \frac{7347439518301}{18868657853079} a^{8} - \frac{1441319713223}{18868657853079} a^{7} - \frac{2929511224910}{6289552617693} a^{6} - \frac{1414257501536}{18868657853079} a^{5} - \frac{43147333193}{2096517539231} a^{4} + \frac{7934216499136}{18868657853079} a^{3} + \frac{2844639614294}{6289552617693} a^{2} - \frac{3888744717355}{18868657853079} a + \frac{3078104076632}{18868657853079}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{2142950200}{23381236497} a^{19} - \frac{6401656598}{23381236497} a^{18} + \frac{5519993277}{7793745499} a^{17} - \frac{26424865711}{23381236497} a^{16} + \frac{18757342914}{7793745499} a^{15} - \frac{106403328202}{23381236497} a^{14} + \frac{50357270021}{7793745499} a^{13} - \frac{176600363578}{23381236497} a^{12} + \frac{223203720991}{23381236497} a^{11} - \frac{123305066234}{7793745499} a^{10} + \frac{157306723888}{7793745499} a^{9} - \frac{401017631845}{23381236497} a^{8} + \frac{430711348906}{23381236497} a^{7} - \frac{199599827819}{7793745499} a^{6} + \frac{603712289920}{23381236497} a^{5} - \frac{139910888471}{7793745499} a^{4} + \frac{244468774555}{23381236497} a^{3} - \frac{114322858484}{7793745499} a^{2} + \frac{279448002566}{23381236497} a - \frac{2862575350}{23381236497} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 110484.915685 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_5\wr C_2$ (as 20T100):

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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 10.6.34424253125.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$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
3319Data not computed