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Properties

Label 20.4.34884167202...8921.2
Degree $20$
Signature $[4, 8]$
Discriminant $11^{18}\cdot 89^{4}$
Root discriminant $21.24$
Ramified primes $11, 89$
Class number $1$
Class group Trivial
Galois group $C_2^2\times C_2^4:C_5$ (as 20T74)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 21, -17, -7, 19, -69, -49, -2, 66, -10, 0, 27, -2, -5, -19, 20, -15, 7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 7*x^18 - 15*x^17 + 20*x^16 - 19*x^15 - 5*x^14 - 2*x^13 + 27*x^12 - 10*x^10 + 66*x^9 - 2*x^8 - 49*x^7 - 69*x^6 + 19*x^5 - 7*x^4 - 17*x^3 + 21*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 + 7*x^18 - 15*x^17 + 20*x^16 - 19*x^15 - 5*x^14 - 2*x^13 + 27*x^12 - 10*x^10 + 66*x^9 - 2*x^8 - 49*x^7 - 69*x^6 + 19*x^5 - 7*x^4 - 17*x^3 + 21*x^2 - 7*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 7 x^{18} - 15 x^{17} + 20 x^{16} - 19 x^{15} - 5 x^{14} - 2 x^{13} + 27 x^{12} - 10 x^{10} + 66 x^{9} - 2 x^{8} - 49 x^{7} - 69 x^{6} + 19 x^{5} - 7 x^{4} - 17 x^{3} + 21 x^{2} - 7 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(348841672023202139208688921=11^{18}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.24$
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:  $11, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{6120963851104427183} a^{19} + \frac{1325031938659587017}{6120963851104427183} a^{18} + \frac{1838599002335896402}{6120963851104427183} a^{17} + \frac{1751414193060666356}{6120963851104427183} a^{16} + \frac{1372952763676366937}{6120963851104427183} a^{15} + \frac{1533300016569738722}{6120963851104427183} a^{14} - \frac{853406164184057415}{6120963851104427183} a^{13} + \frac{490127348076960523}{6120963851104427183} a^{12} - \frac{2887889295721402201}{6120963851104427183} a^{11} - \frac{2183179321194326923}{6120963851104427183} a^{10} + \frac{2251943501850733080}{6120963851104427183} a^{9} + \frac{1714391379652733072}{6120963851104427183} a^{8} + \frac{634114320545558040}{6120963851104427183} a^{7} + \frac{457953227978774054}{6120963851104427183} a^{6} - \frac{975227269709209935}{6120963851104427183} a^{5} + \frac{653529795925459484}{6120963851104427183} a^{4} - \frac{2023018836808056713}{6120963851104427183} a^{3} + \frac{2669278287134501722}{6120963851104427183} a^{2} - \frac{2615109322814242382}{6120963851104427183} a - \frac{2142653373516175997}{6120963851104427183}$

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:  $11$
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:  \( 168465.240789 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T74):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.19077940409.1, 10.4.18677303660411.2, 10.8.209857344499.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
89Data not computed