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Properties

Label 20.0.28211099074...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{38}\cdot 3^{16}\cdot 5^{22}$
Root discriminant $52.79$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1444, 0, 1560, 0, 3860, 0, 1560, 0, 3160, 0, 84, 0, -1340, 0, 210, 0, 200, 0, 30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 30*x^18 + 200*x^16 + 210*x^14 - 1340*x^12 + 84*x^10 + 3160*x^8 + 1560*x^6 + 3860*x^4 + 1560*x^2 + 1444)
 
gp: K = bnfinit(x^20 + 30*x^18 + 200*x^16 + 210*x^14 - 1340*x^12 + 84*x^10 + 3160*x^8 + 1560*x^6 + 3860*x^4 + 1560*x^2 + 1444, 1)
 

Normalized defining polynomial

\( x^{20} + 30 x^{18} + 200 x^{16} + 210 x^{14} - 1340 x^{12} + 84 x^{10} + 3160 x^{8} + 1560 x^{6} + 3860 x^{4} + 1560 x^{2} + 1444 \)

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:  \(28211099074560000000000000000000000=2^{38}\cdot 3^{16}\cdot 5^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.79$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{22} a^{10} + \frac{5}{11}$, $\frac{1}{22} a^{11} + \frac{5}{11} a$, $\frac{1}{22} a^{12} + \frac{5}{11} a^{2}$, $\frac{1}{22} a^{13} + \frac{5}{11} a^{3}$, $\frac{1}{22} a^{14} + \frac{5}{11} a^{4}$, $\frac{1}{22} a^{15} + \frac{5}{11} a^{5}$, $\frac{1}{484} a^{16} - \frac{1}{242} a^{14} + \frac{3}{242} a^{12} + \frac{3}{242} a^{10} - \frac{3}{11} a^{8} - \frac{61}{242} a^{6} - \frac{5}{121} a^{4} - \frac{51}{121} a^{2} + \frac{26}{121}$, $\frac{1}{484} a^{17} - \frac{1}{242} a^{15} + \frac{3}{242} a^{13} + \frac{3}{242} a^{11} - \frac{3}{11} a^{9} - \frac{61}{242} a^{7} - \frac{5}{121} a^{5} - \frac{51}{121} a^{3} + \frac{26}{121} a$, $\frac{1}{232136371490452} a^{18} + \frac{68575524601}{116068185745226} a^{16} - \frac{907672730938}{58034092872613} a^{14} + \frac{505189891507}{116068185745226} a^{12} - \frac{487741656863}{116068185745226} a^{10} + \frac{3973292845893}{8928321980402} a^{8} + \frac{6554646567661}{58034092872613} a^{6} - \frac{7540122177076}{58034092872613} a^{4} + \frac{13868044596184}{58034092872613} a^{2} + \frac{3239270958423}{58034092872613}$, $\frac{1}{4410591058318588} a^{19} + \frac{1576012855961}{4410591058318588} a^{17} - \frac{14816670196275}{1102647764579647} a^{15} + \frac{10097601936567}{2205295529159294} a^{13} + \frac{30207976887329}{2205295529159294} a^{11} + \frac{50238234017067}{169638117627638} a^{9} + \frac{505679651649153}{2205295529159294} a^{7} + \frac{433710831895684}{1102647764579647} a^{5} + \frac{373103875683681}{1102647764579647} a^{3} - \frac{1556935064107}{1102647764579647} a$

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:  $9$
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:  \( 6586884452.706072 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-2}, \sqrt{5})\), 5.1.4050000.3, 10.0.167961600000000000.11, 10.0.33592320000000000.77, 10.2.82012500000000.7

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 10 siblings: 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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$3$3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
5Data not computed