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Properties

Label 20.8.34918889042...5625.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{10}\cdot 11^{10}\cdot 13^{10}$
Root discriminant $26.74$
Ramified primes $5, 11, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-41, 55, 340, -454, -864, 1437, 600, -2186, -262, 1308, 685, -774, -338, 393, -10, -114, 35, 22, -10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 10*x^18 + 22*x^17 + 35*x^16 - 114*x^15 - 10*x^14 + 393*x^13 - 338*x^12 - 774*x^11 + 685*x^10 + 1308*x^9 - 262*x^8 - 2186*x^7 + 600*x^6 + 1437*x^5 - 864*x^4 - 454*x^3 + 340*x^2 + 55*x - 41)
 
gp: K = bnfinit(x^20 - 2*x^19 - 10*x^18 + 22*x^17 + 35*x^16 - 114*x^15 - 10*x^14 + 393*x^13 - 338*x^12 - 774*x^11 + 685*x^10 + 1308*x^9 - 262*x^8 - 2186*x^7 + 600*x^6 + 1437*x^5 - 864*x^4 - 454*x^3 + 340*x^2 + 55*x - 41, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 10 x^{18} + 22 x^{17} + 35 x^{16} - 114 x^{15} - 10 x^{14} + 393 x^{13} - 338 x^{12} - 774 x^{11} + 685 x^{10} + 1308 x^{9} - 262 x^{8} - 2186 x^{7} + 600 x^{6} + 1437 x^{5} - 864 x^{4} - 454 x^{3} + 340 x^{2} + 55 x - 41 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(34918889042392681418447265625=5^{10}\cdot 11^{10}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.74$
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:  $5, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{7} a^{18} + \frac{3}{7} a^{17} - \frac{2}{7} a^{13} + \frac{1}{7} a^{12} + \frac{2}{7} a^{11} + \frac{3}{7} a^{10} + \frac{1}{7} a^{9} + \frac{3}{7} a^{8} + \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} + \frac{3}{7} a^{2} + \frac{3}{7}$, $\frac{1}{646887150435465577852949} a^{19} + \frac{28432038343534129235193}{646887150435465577852949} a^{18} + \frac{24849264630067616505288}{92412450062209368264707} a^{17} + \frac{30895836230842943168209}{92412450062209368264707} a^{16} + \frac{20247754854633236020449}{92412450062209368264707} a^{15} + \frac{11775453393262013246384}{646887150435465577852949} a^{14} + \frac{129714274444440175161792}{646887150435465577852949} a^{13} - \frac{266627947693186257385072}{646887150435465577852949} a^{12} + \frac{173154697134165505703199}{646887150435465577852949} a^{11} + \frac{107974286189152159899676}{646887150435465577852949} a^{10} - \frac{199434597863143131561524}{646887150435465577852949} a^{9} + \frac{70901118319877169806666}{646887150435465577852949} a^{8} + \frac{112042043034638598319402}{646887150435465577852949} a^{7} - \frac{63492166807705527104706}{646887150435465577852949} a^{6} + \frac{71010916790858593674890}{646887150435465577852949} a^{5} + \frac{199877575940177696511635}{646887150435465577852949} a^{4} - \frac{54274894694524929176315}{646887150435465577852949} a^{3} + \frac{40678558271104359403181}{92412450062209368264707} a^{2} - \frac{274502450580214621340951}{646887150435465577852949} a - \frac{17122557174174170487970}{92412450062209368264707}$

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:  $13$
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:  \( 4695779.19315 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.1306755003125.1

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

Sibling fields

Degree 20 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.8.7.2$x^{8} - 11$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.12.8.1$x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$