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Properties

Label 20.0.37493871612...7424.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 11^{10}\cdot 13^{10}$
Root discriminant $23.92$
Ramified primes $2, 11, 13$
Class number $2$
Class group $[2]$
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![121, 0, 253, 0, 564, 0, -1847, 0, 2633, 0, -1523, 0, 613, 0, -211, 0, 60, 0, -11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 11*x^18 + 60*x^16 - 211*x^14 + 613*x^12 - 1523*x^10 + 2633*x^8 - 1847*x^6 + 564*x^4 + 253*x^2 + 121)
 
gp: K = bnfinit(x^20 - 11*x^18 + 60*x^16 - 211*x^14 + 613*x^12 - 1523*x^10 + 2633*x^8 - 1847*x^6 + 564*x^4 + 253*x^2 + 121, 1)
 

Normalized defining polynomial

\( x^{20} - 11 x^{18} + 60 x^{16} - 211 x^{14} + 613 x^{12} - 1523 x^{10} + 2633 x^{8} - 1847 x^{6} + 564 x^{4} + 253 x^{2} + 121 \)

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:  \(3749387161243233107049447424=2^{20}\cdot 11^{10}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.92$
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:  $2, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{55} a^{14} - \frac{5}{11} a^{12} - \frac{24}{55} a^{10} - \frac{3}{55} a^{8} + \frac{16}{55} a^{6} + \frac{17}{55} a^{4} + \frac{1}{11} a^{2} + \frac{2}{5}$, $\frac{1}{55} a^{15} - \frac{5}{11} a^{13} - \frac{24}{55} a^{11} - \frac{3}{55} a^{9} + \frac{16}{55} a^{7} + \frac{17}{55} a^{5} + \frac{1}{11} a^{3} + \frac{2}{5} a$, $\frac{1}{715} a^{16} + \frac{1}{715} a^{14} - \frac{289}{715} a^{12} + \frac{18}{65} a^{10} - \frac{62}{715} a^{8} - \frac{282}{715} a^{6} + \frac{337}{715} a^{4} + \frac{207}{715} a^{2} - \frac{8}{65}$, $\frac{1}{715} a^{17} + \frac{1}{715} a^{15} - \frac{289}{715} a^{13} + \frac{18}{65} a^{11} - \frac{62}{715} a^{9} - \frac{282}{715} a^{7} + \frac{337}{715} a^{5} + \frac{207}{715} a^{3} - \frac{8}{65} a$, $\frac{1}{166647108485} a^{18} + \frac{105866496}{166647108485} a^{16} + \frac{1206325072}{166647108485} a^{14} + \frac{46380643213}{166647108485} a^{12} + \frac{50320773384}{166647108485} a^{10} + \frac{13277393407}{33329421697} a^{8} + \frac{33645679588}{166647108485} a^{6} - \frac{6873420146}{166647108485} a^{4} + \frac{71691417092}{166647108485} a^{2} + \frac{4963336127}{15149737135}$, $\frac{1}{1833118193335} a^{19} + \frac{4727374}{15149737135} a^{17} - \frac{7417371451}{1833118193335} a^{15} - \frac{361030749279}{1833118193335} a^{13} - \frac{139167477243}{1833118193335} a^{11} - \frac{768480085543}{1833118193335} a^{9} - \frac{743186226119}{1833118193335} a^{7} + \frac{328984598493}{1833118193335} a^{5} + \frac{289381486078}{1833118193335} a^{3} - \frac{62394725904}{166647108485} a$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( 314252.606616 \)
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

\(\Q(\sqrt{-143}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{11}, \sqrt{-13})\), 5.1.20449.1 x5, 10.0.59797108943.2, 10.0.5566567232512.1 x5, 10.2.4710172273664.1 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R R ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$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}$