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Properties

Label 20.6.15764000324...6875.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,5^{10}\cdot 19^{9}\cdot 29^{8}$
Root discriminant $32.35$
Ramified primes $5, 19, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T955

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -15, -61, 0, 480, 901, 354, -6, 720, -5, -1465, 1484, -753, 47, 273, -215, 38, 32, -10, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 10*x^18 + 32*x^17 + 38*x^16 - 215*x^15 + 273*x^14 + 47*x^13 - 753*x^12 + 1484*x^11 - 1465*x^10 - 5*x^9 + 720*x^8 - 6*x^7 + 354*x^6 + 901*x^5 + 480*x^4 - 61*x^2 - 15*x - 1)
 
gp: K = bnfinit(x^20 - 3*x^19 - 10*x^18 + 32*x^17 + 38*x^16 - 215*x^15 + 273*x^14 + 47*x^13 - 753*x^12 + 1484*x^11 - 1465*x^10 - 5*x^9 + 720*x^8 - 6*x^7 + 354*x^6 + 901*x^5 + 480*x^4 - 61*x^2 - 15*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 10 x^{18} + 32 x^{17} + 38 x^{16} - 215 x^{15} + 273 x^{14} + 47 x^{13} - 753 x^{12} + 1484 x^{11} - 1465 x^{10} - 5 x^{9} + 720 x^{8} - 6 x^{7} + 354 x^{6} + 901 x^{5} + 480 x^{4} - 61 x^{2} - 15 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1576400032427617153453310546875=-\,5^{10}\cdot 19^{9}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.35$
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:  $5, 19, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3695787673028999684864978954} a^{19} + \frac{151403439205856033224111162}{1847893836514499842432489477} a^{18} - \frac{898043883067449514150843519}{1847893836514499842432489477} a^{17} - \frac{475418347898944104910136121}{1847893836514499842432489477} a^{16} + \frac{56254854125473026407640855}{167990348774045440221135407} a^{15} - \frac{1665017153868795884084578085}{3695787673028999684864978954} a^{14} + \frac{14672827116740489383038606}{167990348774045440221135407} a^{13} + \frac{1709415471453584185167156063}{3695787673028999684864978954} a^{12} - \frac{456498842618890625544936718}{1847893836514499842432489477} a^{11} + \frac{442703373792223690603435578}{1847893836514499842432489477} a^{10} - \frac{5958876464486744171936459}{3695787673028999684864978954} a^{9} - \frac{501552262167709425157022820}{1847893836514499842432489477} a^{8} + \frac{152311054799702897983964797}{1847893836514499842432489477} a^{7} - \frac{634770946696045136155877065}{1847893836514499842432489477} a^{6} + \frac{415130711649746354279918236}{1847893836514499842432489477} a^{5} + \frac{995402454980167745566864507}{3695787673028999684864978954} a^{4} - \frac{1256243656904921821866049623}{3695787673028999684864978954} a^{3} + \frac{409839517616701457780925817}{3695787673028999684864978954} a^{2} + \frac{63217328223469411612877854}{1847893836514499842432489477} a - \frac{1392786199836883275435493485}{3695787673028999684864978954}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $12$
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:  \( 22226716.44 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T955:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 819200
The 275 conjugacy class representatives for t20n955 are not computed
Character table for t20n955 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.9932496465625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $16{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ $16{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ $16{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ $20$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.10.9.1$x^{10} - 19$$10$$1$$9$$D_{10}$$[\ ]_{10}^{2}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$