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Properties

Label 20.0.49008711322...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 419^{2}\cdot 695771^{2}$
Root discriminant $27.20$
Ramified primes $3, 5, 419, 695771$
Class number $15$ (GRH)
Class group $[15]$ (GRH)
Galois group 20T656

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 21, 34, 130, 152, 576, 299, 927, 129, 1048, -43, 676, -163, 320, -91, 95, -32, 19, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 19*x^18 - 32*x^17 + 95*x^16 - 91*x^15 + 320*x^14 - 163*x^13 + 676*x^12 - 43*x^11 + 1048*x^10 + 129*x^9 + 927*x^8 + 299*x^7 + 576*x^6 + 152*x^5 + 130*x^4 + 34*x^3 + 21*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 19*x^18 - 32*x^17 + 95*x^16 - 91*x^15 + 320*x^14 - 163*x^13 + 676*x^12 - 43*x^11 + 1048*x^10 + 129*x^9 + 927*x^8 + 299*x^7 + 576*x^6 + 152*x^5 + 130*x^4 + 34*x^3 + 21*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 19 x^{18} - 32 x^{17} + 95 x^{16} - 91 x^{15} + 320 x^{14} - 163 x^{13} + 676 x^{12} - 43 x^{11} + 1048 x^{10} + 129 x^{9} + 927 x^{8} + 299 x^{7} + 576 x^{6} + 152 x^{5} + 130 x^{4} + 34 x^{3} + 21 x^{2} + 4 x + 1 \)

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:  \(49008711322611047193837890625=3^{10}\cdot 5^{10}\cdot 419^{2}\cdot 695771^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.20$
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:  $3, 5, 419, 695771$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13} a^{18} - \frac{6}{13} a^{17} - \frac{2}{13} a^{16} + \frac{1}{13} a^{15} + \frac{3}{13} a^{14} + \frac{6}{13} a^{11} + \frac{1}{13} a^{10} + \frac{4}{13} a^{9} + \frac{6}{13} a^{8} - \frac{2}{13} a^{7} + \frac{5}{13} a^{6} + \frac{4}{13} a^{5} - \frac{6}{13} a^{3} - \frac{1}{13} a^{2} + \frac{2}{13}$, $\frac{1}{6061522289697179279} a^{19} - \frac{202430534899637212}{6061522289697179279} a^{18} + \frac{465185615638738392}{6061522289697179279} a^{17} - \frac{83011570651921622}{466270945361321483} a^{16} + \frac{2254164603998173825}{6061522289697179279} a^{15} + \frac{1082566193444505713}{6061522289697179279} a^{14} + \frac{49947449980935545}{466270945361321483} a^{13} - \frac{1571058483182505090}{6061522289697179279} a^{12} + \frac{374889206044363183}{6061522289697179279} a^{11} + \frac{108642068982336043}{6061522289697179279} a^{10} - \frac{1209142734672951226}{6061522289697179279} a^{9} - \frac{394153634918993494}{6061522289697179279} a^{8} + \frac{79511223586134071}{6061522289697179279} a^{7} - \frac{91012415978103242}{466270945361321483} a^{6} - \frac{1189575142096405192}{6061522289697179279} a^{5} - \frac{1099010363719159209}{6061522289697179279} a^{4} + \frac{1563007180623317269}{6061522289697179279} a^{3} + \frac{1813935268680334117}{6061522289697179279} a^{2} - \frac{475336632056498483}{6061522289697179279} a - \frac{2497499825697124546}{6061522289697179279}$

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

Class group and class number

$C_{15}$, which has order $15$ (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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1632393088200393874}{6061522289697179279} a^{19} - \frac{6467891859143458924}{6061522289697179279} a^{18} + \frac{30699169188839777865}{6061522289697179279} a^{17} - \frac{50865675637524136632}{6061522289697179279} a^{16} + \frac{152105019082717176514}{6061522289697179279} a^{15} - \frac{10920883947533548362}{466270945361321483} a^{14} + \frac{39439621391469936102}{466270945361321483} a^{13} - \frac{247362926824828144800}{6061522289697179279} a^{12} + \frac{1078429721448063452004}{6061522289697179279} a^{11} - \frac{41434475175582463010}{6061522289697179279} a^{10} + \frac{1674763227994037146730}{6061522289697179279} a^{9} + \frac{229063617564274157579}{6061522289697179279} a^{8} + \frac{1454499658062648406606}{6061522289697179279} a^{7} + \frac{463164264512908457008}{6061522289697179279} a^{6} + \frac{68713882104411684048}{466270945361321483} a^{5} + \frac{205792264809694379546}{6061522289697179279} a^{4} + \frac{169025023830560851014}{6061522289697179279} a^{3} + \frac{1286483868039617338}{466270945361321483} a^{2} + \frac{25680551272583853368}{6061522289697179279} a + \frac{366804933813745930}{466270945361321483} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 161078.71489 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T656:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 70 conjugacy class representatives for t20n656 are not computed
Character table for t20n656 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 10.10.911025153125.1

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 24 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
419Data not computed
695771Data not computed