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Properties

Label 20.0.98418936267...8656.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{24}\cdot 7^{16}$
Root discriminant $35.45$
Ramified primes $2, 3, 7$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 20T145

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7048, -1620, -16823, 6056, 26208, -36688, 25347, -12888, 5907, -3812, 3576, -2696, 1240, -36, -275, 44, 69, -40, 16, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 16*x^18 - 40*x^17 + 69*x^16 + 44*x^15 - 275*x^14 - 36*x^13 + 1240*x^12 - 2696*x^11 + 3576*x^10 - 3812*x^9 + 5907*x^8 - 12888*x^7 + 25347*x^6 - 36688*x^5 + 26208*x^4 + 6056*x^3 - 16823*x^2 - 1620*x + 7048)
 
gp: K = bnfinit(x^20 - 6*x^19 + 16*x^18 - 40*x^17 + 69*x^16 + 44*x^15 - 275*x^14 - 36*x^13 + 1240*x^12 - 2696*x^11 + 3576*x^10 - 3812*x^9 + 5907*x^8 - 12888*x^7 + 25347*x^6 - 36688*x^5 + 26208*x^4 + 6056*x^3 - 16823*x^2 - 1620*x + 7048, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 16 x^{18} - 40 x^{17} + 69 x^{16} + 44 x^{15} - 275 x^{14} - 36 x^{13} + 1240 x^{12} - 2696 x^{11} + 3576 x^{10} - 3812 x^{9} + 5907 x^{8} - 12888 x^{7} + 25347 x^{6} - 36688 x^{5} + 26208 x^{4} + 6056 x^{3} - 16823 x^{2} - 1620 x + 7048 \)

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:  \(9841893626795960030057286598656=2^{20}\cdot 3^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.45$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{50} a^{18} - \frac{23}{50} a^{17} - \frac{9}{50} a^{16} - \frac{19}{50} a^{15} - \frac{7}{25} a^{14} - \frac{7}{25} a^{13} - \frac{13}{50} a^{12} + \frac{9}{50} a^{11} - \frac{1}{10} a^{10} - \frac{1}{10} a^{9} - \frac{9}{50} a^{8} + \frac{21}{50} a^{7} - \frac{3}{25} a^{6} - \frac{11}{25} a^{5} + \frac{17}{50} a^{4} - \frac{1}{2} a^{3} + \frac{11}{50} a^{2} + \frac{19}{50} a - \frac{11}{25}$, $\frac{1}{33114658727038963861874382602012030000} a^{19} - \frac{182552560494806324714014841565708727}{33114658727038963861874382602012030000} a^{18} + \frac{2967917118622935431125472717973513183}{33114658727038963861874382602012030000} a^{17} - \frac{232410756761810416934520694487215691}{2547281440541458758605721738616310000} a^{16} + \frac{3739608363090339799622235898229602703}{8278664681759740965468595650503007500} a^{15} - \frac{168843722511606382515677273334817051}{4139332340879870482734297825251503750} a^{14} + \frac{11568077227908441653632892480544103893}{33114658727038963861874382602012030000} a^{13} + \frac{8132679656974255601899182445348479111}{33114658727038963861874382602012030000} a^{12} - \frac{11838389954072861053393248703959410791}{33114658727038963861874382602012030000} a^{11} + \frac{1394793719646409284230686133315448123}{6622931745407792772374876520402406000} a^{10} + \frac{10890734534615933609841409407467585161}{33114658727038963861874382602012030000} a^{9} - \frac{14877477161476794853099349304322307893}{33114658727038963861874382602012030000} a^{8} + \frac{51234724969059488973317551496156697}{413933234087987048273429782525150375} a^{7} - \frac{653841520483594063472387482777082553}{2069666170439935241367148912625751875} a^{6} - \frac{2161041924942067164163520202655021849}{6622931745407792772374876520402406000} a^{5} - \frac{8955903445602254771795059099841411043}{33114658727038963861874382602012030000} a^{4} - \frac{9511796040832513976632297785541652789}{33114658727038963861874382602012030000} a^{3} - \frac{387150799887013318934444755133653443}{1324586349081558554474975304080481200} a^{2} + \frac{39805496978621043827566768170395351}{636820360135364689651430434654077500} a - \frac{1362611337307283544898957738614502789}{4139332340879870482734297825251503750}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( -\frac{9668050423477325909361}{3592748448391172994590000} a^{19} + \frac{43221287504412349927847}{3592748448391172994590000} a^{18} - \frac{90163814403327067717263}{3592748448391172994590000} a^{17} + \frac{254687174187033567475263}{3592748448391172994590000} a^{16} - \frac{72339605445982605248683}{898187112097793248647500} a^{15} - \frac{103784099614619368943289}{449093556048896624323750} a^{14} + \frac{1360013377083313834026027}{3592748448391172994590000} a^{13} + \frac{2304055130364308658024729}{3592748448391172994590000} a^{12} - \frac{8351958998289067532415849}{3592748448391172994590000} a^{11} + \frac{2727309247904786343346197}{718549689678234598918000} a^{10} - \frac{14736860065084919476834121}{3592748448391172994590000} a^{9} + \frac{16048526150470832842666773}{3592748448391172994590000} a^{8} - \frac{431727037031330674296462}{44909355604889662432375} a^{7} + \frac{4655473392779390412823233}{224546778024448312161875} a^{6} - \frac{27304882043660623441670871}{718549689678234598918000} a^{5} + \frac{156250663121329752702808323}{3592748448391172994590000} a^{4} - \frac{32557308383223031932872171}{3592748448391172994590000} a^{3} - \frac{3585810654861128031063741}{143709937935646919783600} a^{2} + \frac{5115145188353950355841657}{898187112097793248647500} a + \frac{5560244889818707672428229}{449093556048896624323750} \) (order $4$)
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:  \( 78497371.7756 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T145:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 720
The 11 conjugacy class representatives for t20n145
Character table for t20n145

Intermediate fields

\(\Q(\sqrt{-1}) \), 10.4.196073702927424.1

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

Sibling fields

Degree 6 siblings: 6.0.12446784.1, 6.4.63011844.2
Degree 10 sibling: 10.4.196073702927424.1
Degree 12 siblings: Deg 12, Deg 12
Degree 15 siblings: Deg 15, Deg 15
Degree 20 siblings: Deg 20, Deg 20
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 sibling: 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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
3Data not computed
$7$7.10.8.1$x^{10} - 7 x^{5} + 147$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
7.10.8.1$x^{10} - 7 x^{5} + 147$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$