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Properties

Label 20.4.80642417151...8961.1
Degree $20$
Signature $[4, 8]$
Discriminant $7^{12}\cdot 17^{12}$
Root discriminant $17.59$
Ramified primes $7, 17$
Class number $1$
Class group Trivial
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![-19, 135, -350, 248, 352, -441, -365, 693, -22, -468, 141, 289, -180, -119, 119, 36, -70, 18, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 9*x^18 + 18*x^17 - 70*x^16 + 36*x^15 + 119*x^14 - 119*x^13 - 180*x^12 + 289*x^11 + 141*x^10 - 468*x^9 - 22*x^8 + 693*x^7 - 365*x^6 - 441*x^5 + 352*x^4 + 248*x^3 - 350*x^2 + 135*x - 19)
 
gp: K = bnfinit(x^20 - 6*x^19 + 9*x^18 + 18*x^17 - 70*x^16 + 36*x^15 + 119*x^14 - 119*x^13 - 180*x^12 + 289*x^11 + 141*x^10 - 468*x^9 - 22*x^8 + 693*x^7 - 365*x^6 - 441*x^5 + 352*x^4 + 248*x^3 - 350*x^2 + 135*x - 19, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 9 x^{18} + 18 x^{17} - 70 x^{16} + 36 x^{15} + 119 x^{14} - 119 x^{13} - 180 x^{12} + 289 x^{11} + 141 x^{10} - 468 x^{9} - 22 x^{8} + 693 x^{7} - 365 x^{6} - 441 x^{5} + 352 x^{4} + 248 x^{3} - 350 x^{2} + 135 x - 19 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{11286437581085631677} a^{19} + \frac{4511743211367842981}{11286437581085631677} a^{18} - \frac{3722333139470782633}{11286437581085631677} a^{17} - \frac{593181573380308717}{11286437581085631677} a^{16} + \frac{869444726534732605}{11286437581085631677} a^{15} - \frac{3848500877478491529}{11286437581085631677} a^{14} - \frac{3513709675537980699}{11286437581085631677} a^{13} + \frac{392793888733804304}{11286437581085631677} a^{12} + \frac{2125325086495588869}{11286437581085631677} a^{11} + \frac{4777941317005806607}{11286437581085631677} a^{10} + \frac{4393366603167836598}{11286437581085631677} a^{9} - \frac{3952455064386546497}{11286437581085631677} a^{8} + \frac{2309875477877149727}{11286437581085631677} a^{7} + \frac{5003847028954449934}{11286437581085631677} a^{6} - \frac{1390366174659717618}{11286437581085631677} a^{5} - \frac{4879249213543569628}{11286437581085631677} a^{4} - \frac{4467433171520039530}{11286437581085631677} a^{3} + \frac{953058539233127990}{11286437581085631677} a^{2} + \frac{433576676819278554}{11286437581085631677} a - \frac{3053668512415094818}{11286437581085631677}$

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:  \( 30036.2093459 \)
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, 10.2.3409076657.1, 10.2.167044756193.1, 10.2.2839760855281.1

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 ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$7.4.2.1$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$$[\ ]_{2}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$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}$