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Properties

Label 20.20.1420748158...3856.2
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 11^{16}\cdot 7369^{4}$
Root discriminant $80.84$
Ramified primes $2, 11, 7369$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T254

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![54302161, 0, -267981054, 0, 392014338, 0, -264094303, 0, 94432719, 0, -19074528, 0, 2232986, 0, -152482, 0, 5932, 0, -121, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 121*x^18 + 5932*x^16 - 152482*x^14 + 2232986*x^12 - 19074528*x^10 + 94432719*x^8 - 264094303*x^6 + 392014338*x^4 - 267981054*x^2 + 54302161)
 
gp: K = bnfinit(x^20 - 121*x^18 + 5932*x^16 - 152482*x^14 + 2232986*x^12 - 19074528*x^10 + 94432719*x^8 - 264094303*x^6 + 392014338*x^4 - 267981054*x^2 + 54302161, 1)
 

Normalized defining polynomial

\( x^{20} - 121 x^{18} + 5932 x^{16} - 152482 x^{14} + 2232986 x^{12} - 19074528 x^{10} + 94432719 x^{8} - 264094303 x^{6} + 392014338 x^{4} - 267981054 x^{2} + 54302161 \)

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:  \(142074815881149027354713197399801593856=2^{20}\cdot 11^{16}\cdot 7369^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.84$
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:  $2, 11, 7369$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{8} - \frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{9} - \frac{1}{5} a^{5} + \frac{2}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{25} a^{14} + \frac{1}{25} a^{12} + \frac{1}{25} a^{10} + \frac{11}{25} a^{8} - \frac{1}{25} a^{6} - \frac{9}{25} a^{4} - \frac{12}{25} a^{2} + \frac{11}{25}$, $\frac{1}{25} a^{15} + \frac{1}{25} a^{13} + \frac{1}{25} a^{11} + \frac{11}{25} a^{9} - \frac{1}{25} a^{7} - \frac{9}{25} a^{5} - \frac{12}{25} a^{3} + \frac{11}{25} a$, $\frac{1}{125} a^{16} + \frac{2}{125} a^{14} + \frac{2}{125} a^{12} + \frac{37}{125} a^{10} + \frac{12}{25} a^{8} - \frac{7}{25} a^{6} - \frac{21}{125} a^{4} + \frac{24}{125} a^{2} - \frac{14}{125}$, $\frac{1}{125} a^{17} + \frac{2}{125} a^{15} + \frac{2}{125} a^{13} + \frac{37}{125} a^{11} + \frac{12}{25} a^{9} - \frac{7}{25} a^{7} - \frac{21}{125} a^{5} + \frac{24}{125} a^{3} - \frac{14}{125} a$, $\frac{1}{30761674194352989165474257710625} a^{18} + \frac{34716953661664462289545831753}{30761674194352989165474257710625} a^{16} - \frac{311537632674293807187196023371}{30761674194352989165474257710625} a^{14} - \frac{1513252619899973197958648005086}{30761674194352989165474257710625} a^{12} - \frac{11623878804973517654703985340403}{30761674194352989165474257710625} a^{10} + \frac{422377639206352188652459188026}{1230466967774119566618970308425} a^{8} - \frac{10975510994194553568367573659931}{30761674194352989165474257710625} a^{6} - \frac{8328296198652642296268777028622}{30761674194352989165474257710625} a^{4} + \frac{1321771227451673256607008076752}{6152334838870597833094851542125} a^{2} + \frac{453370234748248893491504169}{4174470646539963246773545625}$, $\frac{1}{30761674194352989165474257710625} a^{19} + \frac{34716953661664462289545831753}{30761674194352989165474257710625} a^{17} - \frac{311537632674293807187196023371}{30761674194352989165474257710625} a^{15} - \frac{1513252619899973197958648005086}{30761674194352989165474257710625} a^{13} - \frac{11623878804973517654703985340403}{30761674194352989165474257710625} a^{11} + \frac{422377639206352188652459188026}{1230466967774119566618970308425} a^{9} - \frac{10975510994194553568367573659931}{30761674194352989165474257710625} a^{7} - \frac{8328296198652642296268777028622}{30761674194352989165474257710625} a^{5} + \frac{1321771227451673256607008076752}{6152334838870597833094851542125} a^{3} + \frac{453370234748248893491504169}{4174470646539963246773545625} a$

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:  \( 4744435151830 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T254:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2560
The 28 conjugacy class representatives for t20n254
Character table for t20n254 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.1579610594089.1

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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.10.10.3$x^{10} - 9 x^{8} + 22 x^{6} - 46 x^{4} + 9 x^{2} - 9$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
2.10.10.3$x^{10} - 9 x^{8} + 22 x^{6} - 46 x^{4} + 9 x^{2} - 9$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
7369Data not computed