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Properties

Label 20.0.11273107267...2576.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 401^{10}$
Root discriminant $40.05$
Ramified primes $2, 401$
Class number $250$ (GRH)
Class group $[5, 50]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 44, 0, 546, 0, 2906, 0, 7573, 0, 10341, 0, 7573, 0, 2906, 0, 546, 0, 44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 44*x^18 + 546*x^16 + 2906*x^14 + 7573*x^12 + 10341*x^10 + 7573*x^8 + 2906*x^6 + 546*x^4 + 44*x^2 + 1)
 
gp: K = bnfinit(x^20 + 44*x^18 + 546*x^16 + 2906*x^14 + 7573*x^12 + 10341*x^10 + 7573*x^8 + 2906*x^6 + 546*x^4 + 44*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 44 x^{18} + 546 x^{16} + 2906 x^{14} + 7573 x^{12} + 10341 x^{10} + 7573 x^{8} + 2906 x^{6} + 546 x^{4} + 44 x^{2} + 1 \)

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:  \(112731072674656014761673622552576=2^{20}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.05$
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:  $2, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{10} - \frac{1}{9} a^{6} - \frac{2}{9} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{11} - \frac{1}{9} a^{7} - \frac{2}{9} a^{3} + \frac{1}{9} a$, $\frac{1}{27} a^{14} - \frac{4}{27} a^{10} - \frac{4}{27} a^{8} + \frac{4}{27} a^{6} - \frac{5}{27} a^{4} + \frac{1}{3} a^{2} - \frac{1}{27}$, $\frac{1}{27} a^{15} - \frac{4}{27} a^{11} - \frac{4}{27} a^{9} + \frac{4}{27} a^{7} - \frac{5}{27} a^{5} + \frac{1}{3} a^{3} - \frac{1}{27} a$, $\frac{1}{4293} a^{16} - \frac{25}{4293} a^{14} + \frac{203}{4293} a^{12} + \frac{230}{1431} a^{10} + \frac{464}{4293} a^{8} - \frac{35}{1431} a^{6} - \frac{910}{4293} a^{4} - \frac{820}{4293} a^{2} - \frac{794}{4293}$, $\frac{1}{4293} a^{17} - \frac{25}{4293} a^{15} + \frac{203}{4293} a^{13} + \frac{230}{1431} a^{11} + \frac{464}{4293} a^{9} - \frac{35}{1431} a^{7} - \frac{910}{4293} a^{5} - \frac{820}{4293} a^{3} - \frac{794}{4293} a$, $\frac{1}{4293} a^{18} + \frac{55}{4293} a^{14} + \frac{41}{4293} a^{12} + \frac{65}{4293} a^{10} - \frac{430}{4293} a^{8} - \frac{196}{4293} a^{6} - \frac{197}{4293} a^{4} + \frac{19}{477} a^{2} + \frac{1138}{4293}$, $\frac{1}{4293} a^{19} + \frac{55}{4293} a^{15} + \frac{41}{4293} a^{13} + \frac{65}{4293} a^{11} - \frac{430}{4293} a^{9} - \frac{196}{4293} a^{7} - \frac{197}{4293} a^{5} + \frac{19}{477} a^{3} + \frac{1138}{4293} a$

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

Class group and class number

$C_{5}\times C_{50}$, which has order $250$ (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:  \( \frac{229}{477} a^{19} + \frac{3358}{159} a^{17} + \frac{124961}{477} a^{15} + \frac{221672}{159} a^{13} + \frac{1735481}{477} a^{11} + \frac{2383627}{477} a^{9} + \frac{1766660}{477} a^{7} + \frac{680474}{477} a^{5} + \frac{117499}{477} a^{3} + \frac{5941}{477} a \) (order $4$)
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:  \( 795087.603907 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{401}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-401}) \), \(\Q(i, \sqrt{401})\), 5.5.160801.1 x5, 10.10.10368641602001.1, 10.0.26477528679424.1 x5, 10.0.10617489000449024.3 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
401Data not computed