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Properties

Label 20.0.38028917784...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 7^{10}\cdot 13^{10}$
Root discriminant $21.33$
Ramified primes $5, 7, 13$
Class number $4$
Class group $[4]$
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![49, -49, -371, 910, 779, -6851, 15762, -21290, 18621, -9513, 269, 4350, -4378, 2386, -632, -140, 247, -138, 47, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 47*x^18 - 138*x^17 + 247*x^16 - 140*x^15 - 632*x^14 + 2386*x^13 - 4378*x^12 + 4350*x^11 + 269*x^10 - 9513*x^9 + 18621*x^8 - 21290*x^7 + 15762*x^6 - 6851*x^5 + 779*x^4 + 910*x^3 - 371*x^2 - 49*x + 49)
 
gp: K = bnfinit(x^20 - 10*x^19 + 47*x^18 - 138*x^17 + 247*x^16 - 140*x^15 - 632*x^14 + 2386*x^13 - 4378*x^12 + 4350*x^11 + 269*x^10 - 9513*x^9 + 18621*x^8 - 21290*x^7 + 15762*x^6 - 6851*x^5 + 779*x^4 + 910*x^3 - 371*x^2 - 49*x + 49, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 47 x^{18} - 138 x^{17} + 247 x^{16} - 140 x^{15} - 632 x^{14} + 2386 x^{13} - 4378 x^{12} + 4350 x^{11} + 269 x^{10} - 9513 x^{9} + 18621 x^{8} - 21290 x^{7} + 15762 x^{6} - 6851 x^{5} + 779 x^{4} + 910 x^{3} - 371 x^{2} - 49 x + 49 \)

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:  \(380289177849714310556640625=5^{10}\cdot 7^{10}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.33$
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:  $5, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} - \frac{2}{7} a^{8} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{15} - \frac{2}{7} a^{9} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{16} - \frac{2}{7} a^{10} + \frac{1}{7} a^{4}$, $\frac{1}{7} a^{17} - \frac{2}{7} a^{11} + \frac{1}{7} a^{5}$, $\frac{1}{218971981459} a^{18} - \frac{9}{218971981459} a^{17} - \frac{8291536551}{218971981459} a^{16} + \frac{3768869338}{218971981459} a^{15} - \frac{4456982428}{218971981459} a^{14} - \frac{4965191019}{31281711637} a^{13} + \frac{2354768898}{9520520933} a^{12} + \frac{79385850877}{218971981459} a^{11} + \frac{17910279479}{218971981459} a^{10} - \frac{82281220195}{218971981459} a^{9} + \frac{52990756692}{218971981459} a^{8} + \frac{14122292837}{31281711637} a^{7} - \frac{5023529097}{218971981459} a^{6} - \frac{3830575904}{9520520933} a^{5} + \frac{1803369140}{19906543769} a^{4} - \frac{12100621789}{218971981459} a^{3} + \frac{63459753473}{218971981459} a^{2} + \frac{9088164220}{31281711637} a + \frac{2810514490}{31281711637}$, $\frac{1}{127222721227679} a^{19} + \frac{281}{127222721227679} a^{18} - \frac{7953846294959}{127222721227679} a^{17} - \frac{7499695727283}{127222721227679} a^{16} + \frac{1160985749131}{18174674461097} a^{15} - \frac{87519705902}{1652243132827} a^{14} + \frac{15375571765328}{127222721227679} a^{13} - \frac{26256926039591}{127222721227679} a^{12} - \frac{51504511797162}{127222721227679} a^{11} - \frac{63082431539945}{127222721227679} a^{10} + \frac{3409252204990}{18174674461097} a^{9} - \frac{6942681420291}{18174674461097} a^{8} + \frac{26911455078341}{127222721227679} a^{7} + \frac{4148344834012}{127222721227679} a^{6} + \frac{22581168278566}{127222721227679} a^{5} - \frac{21912386152297}{127222721227679} a^{4} - \frac{7651288943973}{18174674461097} a^{3} - \frac{3890805006421}{18174674461097} a^{2} + \frac{1007014211301}{2596382065871} a + \frac{491816139944}{2596382065871}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( 74813.5762693 \)
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

\(\Q(\sqrt{-455}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-7}, \sqrt{65})\), 5.1.207025.1 x5, 10.0.19501004534375.1, 10.0.300015454375.1 x5, 10.2.2785857790625.1 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$