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Properties

Label 20.0.59699157574...1344.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{18}$
Root discriminant $24.48$
Ramified primes $2, 11$
Class number $5$
Class group $[5]$
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![100, 0, 278, 0, 9, 0, -116, 0, 301, 0, -10, 0, 300, 0, 108, 0, 49, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 4*x^18 + 49*x^16 + 108*x^14 + 300*x^12 - 10*x^10 + 301*x^8 - 116*x^6 + 9*x^4 + 278*x^2 + 100)
 
gp: K = bnfinit(x^20 + 4*x^18 + 49*x^16 + 108*x^14 + 300*x^12 - 10*x^10 + 301*x^8 - 116*x^6 + 9*x^4 + 278*x^2 + 100, 1)
 

Normalized defining polynomial

\( x^{20} + 4 x^{18} + 49 x^{16} + 108 x^{14} + 300 x^{12} - 10 x^{10} + 301 x^{8} - 116 x^{6} + 9 x^{4} + 278 x^{2} + 100 \)

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:  \(5969915757478328440239161344=2^{30}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.48$
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:  $2, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{6}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{11} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{12} - \frac{1}{8} a^{9} + \frac{1}{8} a^{6}$, $\frac{1}{224} a^{16} + \frac{3}{112} a^{14} + \frac{5}{56} a^{12} + \frac{1}{16} a^{10} + \frac{3}{56} a^{8} - \frac{1}{4} a^{6} - \frac{79}{224} a^{4} - \frac{1}{2} a^{3} + \frac{31}{112} a^{2} + \frac{17}{56}$, $\frac{1}{2240} a^{17} - \frac{1}{448} a^{16} + \frac{3}{1120} a^{15} - \frac{3}{224} a^{14} + \frac{19}{560} a^{13} + \frac{9}{112} a^{12} - \frac{3}{32} a^{11} - \frac{1}{32} a^{10} + \frac{23}{112} a^{9} - \frac{3}{112} a^{8} - \frac{1}{4} a^{6} - \frac{79}{2240} a^{5} - \frac{145}{448} a^{4} + \frac{143}{1120} a^{3} + \frac{81}{224} a^{2} - \frac{151}{560} a + \frac{39}{112}$, $\frac{1}{4500684160} a^{18} - \frac{6921039}{4500684160} a^{16} + \frac{13504423}{2250342080} a^{14} + \frac{1493907}{19568192} a^{12} - \frac{1}{8} a^{11} - \frac{50979953}{450068416} a^{10} - \frac{1}{4} a^{9} - \frac{3563395}{225034208} a^{8} + \frac{783174481}{4500684160} a^{6} + \frac{1}{8} a^{5} + \frac{1206539721}{4500684160} a^{4} - \frac{1}{4} a^{3} + \frac{764709783}{2250342080} a^{2} - \frac{1}{2} a + \frac{93641335}{225034208}$, $\frac{1}{9001368320} a^{19} - \frac{1}{9001368320} a^{18} + \frac{1115897}{9001368320} a^{17} + \frac{6921039}{9001368320} a^{16} - \frac{243677529}{4500684160} a^{15} + \frac{267788337}{4500684160} a^{14} - \frac{530327}{27954560} a^{13} - \frac{1493907}{39136384} a^{12} + \frac{5278599}{900136832} a^{11} - \frac{61537151}{900136832} a^{10} + \frac{13842615}{64295488} a^{9} - \frac{24565881}{450068416} a^{8} - \frac{341996559}{9001368320} a^{7} + \frac{1467167599}{9001368320} a^{6} + \frac{571621777}{9001368320} a^{5} + \frac{2168973399}{9001368320} a^{4} + \frac{788820591}{4500684160} a^{3} + \frac{1485632297}{4500684160} a^{2} - \frac{106481523}{321477440} a + \frac{131392873}{450068416}$

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

Class group and class number

$C_{5}$, which has order $5$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 331105.28715 \)
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

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-2}, \sqrt{-11})\), 5.1.937024.1 x5, 10.0.7024111812608.2, 10.2.77265229938688.2 x5, 10.0.9658153742336.1 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$11$11.10.9.6$x^{10} + 216513$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.6$x^{10} + 216513$$10$$1$$9$$C_{10}$$[\ ]_{10}$