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Properties

Label 20.0.68625588392...7801.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 7^{8}\cdot 17^{10}$
Root discriminant $15.55$
Ramified primes $3, 7, 17$
Class number $1$
Class group Trivial
Galois group $C_2^2\times D_5$ (as 20T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 11, -4, -16, 25, 1, -26, 30, -9, -13, 24, -10, -9, 17, -5, -5, 6, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - x^18 + 6*x^17 - 5*x^16 - 5*x^15 + 17*x^14 - 9*x^13 - 10*x^12 + 24*x^11 - 13*x^10 - 9*x^9 + 30*x^8 - 26*x^7 + x^6 + 25*x^5 - 16*x^4 - 4*x^3 + 11*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^20 - x^19 - x^18 + 6*x^17 - 5*x^16 - 5*x^15 + 17*x^14 - 9*x^13 - 10*x^12 + 24*x^11 - 13*x^10 - 9*x^9 + 30*x^8 - 26*x^7 + x^6 + 25*x^5 - 16*x^4 - 4*x^3 + 11*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - x^{18} + 6 x^{17} - 5 x^{16} - 5 x^{15} + 17 x^{14} - 9 x^{13} - 10 x^{12} + 24 x^{11} - 13 x^{10} - 9 x^{9} + 30 x^{8} - 26 x^{7} + x^{6} + 25 x^{5} - 16 x^{4} - 4 x^{3} + 11 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(686255883923847255777801=3^{10}\cdot 7^{8}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.55$
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:  $3, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{4199} a^{18} - \frac{1226}{4199} a^{17} + \frac{1110}{4199} a^{16} + \frac{1523}{4199} a^{15} + \frac{1467}{4199} a^{14} - \frac{531}{4199} a^{13} + \frac{1622}{4199} a^{12} + \frac{1158}{4199} a^{11} + \frac{135}{4199} a^{10} - \frac{426}{4199} a^{9} - \frac{81}{323} a^{8} + \frac{1091}{4199} a^{7} + \frac{150}{4199} a^{6} - \frac{1796}{4199} a^{5} + \frac{92}{247} a^{4} + \frac{610}{4199} a^{3} + \frac{1380}{4199} a^{2} + \frac{87}{4199} a + \frac{973}{4199}$, $\frac{1}{348517} a^{19} - \frac{6}{348517} a^{18} + \frac{9062}{26809} a^{17} - \frac{554}{348517} a^{16} + \frac{41360}{348517} a^{15} + \frac{29828}{348517} a^{14} + \frac{113821}{348517} a^{13} - \frac{27124}{348517} a^{12} - \frac{165929}{348517} a^{11} - \frac{5277}{18343} a^{10} + \frac{20898}{348517} a^{9} - \frac{91053}{348517} a^{8} + \frac{159649}{348517} a^{7} + \frac{139214}{348517} a^{6} - \frac{106852}{348517} a^{5} - \frac{140422}{348517} a^{4} - \frac{157205}{348517} a^{3} + \frac{138455}{348517} a^{2} + \frac{128108}{348517} a - \frac{152421}{348517}$

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:  \( -\frac{5818}{18343} a^{19} + \frac{9759}{18343} a^{18} + \frac{7321}{18343} a^{17} - \frac{39226}{18343} a^{16} + \frac{44280}{18343} a^{15} + \frac{34695}{18343} a^{14} - \frac{7034}{1079} a^{13} + \frac{79129}{18343} a^{12} + \frac{80689}{18343} a^{11} - \frac{162356}{18343} a^{10} + \frac{104060}{18343} a^{9} + \frac{68491}{18343} a^{8} - \frac{201595}{18343} a^{7} + \frac{196194}{18343} a^{6} - \frac{8882}{18343} a^{5} - \frac{174412}{18343} a^{4} + \frac{122283}{18343} a^{3} + \frac{56020}{18343} a^{2} - \frac{79743}{18343} a + \frac{29314}{18343} \) (order $6$)
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:  \( 12584.3597894 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times D_5$ (as 20T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 16 conjugacy class representatives for $C_2^2\times D_5$
Character table for $C_2^2\times D_5$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-3}, \sqrt{17})\), 5.1.14161.1, 10.0.828405627651.1, 10.0.48729742803.1, 10.2.3409076657.1

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

Sibling fields

Galois closure: data not computed
Degree 20 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
3Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$