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Properties

Label 20.0.10912062142...4801.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{10}\cdot 29^{10}$
Root discriminant $17.86$
Ramified primes $11, 29$
Class number $1$
Class group Trivial
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![9, -60, 205, -179, 382, -162, 936, -276, 111, -546, 397, -125, 139, -135, 45, -30, 24, -6, 1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + x^18 - 6*x^17 + 24*x^16 - 30*x^15 + 45*x^14 - 135*x^13 + 139*x^12 - 125*x^11 + 397*x^10 - 546*x^9 + 111*x^8 - 276*x^7 + 936*x^6 - 162*x^5 + 382*x^4 - 179*x^3 + 205*x^2 - 60*x + 9)
 
gp: K = bnfinit(x^20 - 2*x^19 + x^18 - 6*x^17 + 24*x^16 - 30*x^15 + 45*x^14 - 135*x^13 + 139*x^12 - 125*x^11 + 397*x^10 - 546*x^9 + 111*x^8 - 276*x^7 + 936*x^6 - 162*x^5 + 382*x^4 - 179*x^3 + 205*x^2 - 60*x + 9, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + x^{18} - 6 x^{17} + 24 x^{16} - 30 x^{15} + 45 x^{14} - 135 x^{13} + 139 x^{12} - 125 x^{11} + 397 x^{10} - 546 x^{9} + 111 x^{8} - 276 x^{7} + 936 x^{6} - 162 x^{5} + 382 x^{4} - 179 x^{3} + 205 x^{2} - 60 x + 9 \)

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:  \(10912062142819279835644801=11^{10}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.86$
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:  $11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{8}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} + \frac{1}{3} a^{8} - \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{27} a^{18} - \frac{1}{27} a^{17} - \frac{1}{9} a^{16} + \frac{1}{9} a^{15} - \frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{27} a^{10} + \frac{2}{27} a^{9} + \frac{1}{3} a^{8} - \frac{4}{9} a^{7} - \frac{2}{9} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{7}{27} a^{2} - \frac{1}{27} a + \frac{1}{9}$, $\frac{1}{34156820388509920969929} a^{19} - \frac{204418070098710911593}{34156820388509920969929} a^{18} + \frac{103151448068763775745}{11385606796169973656643} a^{17} - \frac{244947032694873937175}{11385606796169973656643} a^{16} - \frac{14602687674011089921}{421689140598887913209} a^{15} + \frac{1406301233666866317221}{11385606796169973656643} a^{14} + \frac{1289475328766258547935}{11385606796169973656643} a^{13} + \frac{645605088271136429561}{11385606796169973656643} a^{12} + \frac{5281226455286265174847}{34156820388509920969929} a^{11} - \frac{3995554225260263528992}{34156820388509920969929} a^{10} + \frac{127435407760161453572}{1265067421796663739627} a^{9} - \frac{36391397289730193176}{11385606796169973656643} a^{8} + \frac{326467500612650071807}{1265067421796663739627} a^{7} - \frac{2861391991255825910372}{11385606796169973656643} a^{6} - \frac{4358315628839674470914}{11385606796169973656643} a^{5} + \frac{5465292421029167802925}{11385606796169973656643} a^{4} - \frac{11318297044758259452824}{34156820388509920969929} a^{3} - \frac{7377626251606850128159}{34156820388509920969929} a^{2} + \frac{4167694490484903071095}{11385606796169973656643} a - \frac{93437044625382900685}{1265067421796663739627}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( 30662.8930322 \)
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

\(\Q(\sqrt{-319}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-11}, \sqrt{29})\), 5.1.101761.1 x5, 10.0.3303341057599.3, 10.2.300303732509.1 x5, 10.0.113908312331.3 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$