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Properties

Label 20.4.80642417151...8961.2
Degree $20$
Signature $[4, 8]$
Discriminant $7^{12}\cdot 17^{12}$
Root discriminant $17.59$
Ramified primes $7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T73)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-32, 16, 192, -148, -426, 427, 392, -537, -53, 291, -198, 17, 160, -93, -42, 33, -1, 3, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 3*x^17 - x^16 + 33*x^15 - 42*x^14 - 93*x^13 + 160*x^12 + 17*x^11 - 198*x^10 + 291*x^9 - 53*x^8 - 537*x^7 + 392*x^6 + 427*x^5 - 426*x^4 - 148*x^3 + 192*x^2 + 16*x - 32)
 
gp: K = bnfinit(x^20 - 3*x^19 + 3*x^17 - x^16 + 33*x^15 - 42*x^14 - 93*x^13 + 160*x^12 + 17*x^11 - 198*x^10 + 291*x^9 - 53*x^8 - 537*x^7 + 392*x^6 + 427*x^5 - 426*x^4 - 148*x^3 + 192*x^2 + 16*x - 32, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 3 x^{17} - x^{16} + 33 x^{15} - 42 x^{14} - 93 x^{13} + 160 x^{12} + 17 x^{11} - 198 x^{10} + 291 x^{9} - 53 x^{8} - 537 x^{7} + 392 x^{6} + 427 x^{5} - 426 x^{4} - 148 x^{3} + 192 x^{2} + 16 x - 32 \)

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:  \(8064241715186276625588961=7^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.59$
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:  $7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{14} a^{16} + \frac{5}{14} a^{15} + \frac{1}{7} a^{14} + \frac{1}{14} a^{13} + \frac{5}{14} a^{12} + \frac{5}{14} a^{11} + \frac{2}{7} a^{10} + \frac{5}{14} a^{9} + \frac{1}{7} a^{8} + \frac{3}{14} a^{7} - \frac{1}{2} a^{5} - \frac{3}{14} a^{4} - \frac{1}{2} a^{3} + \frac{1}{7} a^{2} - \frac{3}{14} a - \frac{1}{7}$, $\frac{1}{28} a^{17} - \frac{1}{28} a^{16} - \frac{1}{2} a^{15} + \frac{3}{28} a^{14} + \frac{13}{28} a^{13} + \frac{3}{28} a^{12} - \frac{3}{7} a^{11} - \frac{5}{28} a^{10} - \frac{1}{2} a^{9} + \frac{5}{28} a^{8} - \frac{1}{7} a^{7} + \frac{1}{4} a^{6} - \frac{3}{28} a^{5} - \frac{3}{28} a^{4} + \frac{1}{14} a^{3} - \frac{1}{28} a^{2} - \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{56} a^{18} - \frac{1}{56} a^{17} - \frac{1}{28} a^{16} + \frac{1}{8} a^{15} - \frac{19}{56} a^{14} - \frac{13}{56} a^{13} + \frac{5}{14} a^{12} + \frac{27}{56} a^{11} + \frac{3}{28} a^{10} - \frac{19}{56} a^{9} + \frac{5}{14} a^{8} - \frac{13}{56} a^{7} + \frac{25}{56} a^{6} + \frac{25}{56} a^{5} - \frac{3}{28} a^{4} - \frac{1}{56} a^{3} + \frac{3}{14} a^{2} + \frac{1}{14} a - \frac{3}{7}$, $\frac{1}{2593115728} a^{19} - \frac{3838521}{2593115728} a^{18} - \frac{18418585}{1296557864} a^{17} - \frac{52640601}{2593115728} a^{16} + \frac{1130712101}{2593115728} a^{15} - \frac{154997}{2593115728} a^{14} - \frac{18990367}{92611276} a^{13} - \frac{181185707}{370445104} a^{12} + \frac{387321111}{1296557864} a^{11} - \frac{650495451}{2593115728} a^{10} - \frac{197684421}{648278932} a^{9} + \frac{536655955}{2593115728} a^{8} + \frac{552791481}{2593115728} a^{7} - \frac{1052039159}{2593115728} a^{6} - \frac{142299315}{1296557864} a^{5} + \frac{464678983}{2593115728} a^{4} + \frac{114594607}{648278932} a^{3} - \frac{138254077}{648278932} a^{2} - \frac{140001899}{324139466} a + \frac{383608}{23152819}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 31034.6632179 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T73):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

\(\Q(\sqrt{17}) \), 5.1.14161.1, 10.2.167044756193.2, 10.2.3409076657.1, 10.2.2839760855281.2

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
Degree 20 siblings: data not computed
Degree 32 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$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}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{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$$[\ ]_{2}^{2}$
17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$