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Properties

Label 20.0.39848352687...4777.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 17^{4}\cdot 97^{7}$
Root discriminant $15.14$
Ramified primes $3, 17, 97$
Class number $1$
Class group Trivial
Galois group 20T174

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 0, 30, 177, 139, -179, -322, -17, 301, 111, -164, -101, 63, 60, -18, -26, 4, 8, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 8*x^17 + 4*x^16 - 26*x^15 - 18*x^14 + 60*x^13 + 63*x^12 - 101*x^11 - 164*x^10 + 111*x^9 + 301*x^8 - 17*x^7 - 322*x^6 - 179*x^5 + 139*x^4 + 177*x^3 + 30*x^2 + 9)
 
gp: K = bnfinit(x^20 - 3*x^19 + 8*x^17 + 4*x^16 - 26*x^15 - 18*x^14 + 60*x^13 + 63*x^12 - 101*x^11 - 164*x^10 + 111*x^9 + 301*x^8 - 17*x^7 - 322*x^6 - 179*x^5 + 139*x^4 + 177*x^3 + 30*x^2 + 9, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 8 x^{17} + 4 x^{16} - 26 x^{15} - 18 x^{14} + 60 x^{13} + 63 x^{12} - 101 x^{11} - 164 x^{10} + 111 x^{9} + 301 x^{8} - 17 x^{7} - 322 x^{6} - 179 x^{5} + 139 x^{4} + 177 x^{3} + 30 x^{2} + 9 \)

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:  \(398483526878269003824777=3^{10}\cdot 17^{4}\cdot 97^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.14$
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:  $3, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{9} a^{18} + \frac{1}{3} a^{16} - \frac{1}{9} a^{15} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} - \frac{1}{3} a^{12} - \frac{2}{9} a^{9} + \frac{1}{9} a^{8} - \frac{2}{9} a^{6} + \frac{4}{9} a^{5} - \frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{125718677761379967} a^{19} - \frac{3110033823957463}{125718677761379967} a^{18} + \frac{20809834703503936}{41906225920459989} a^{17} + \frac{60926169326681885}{125718677761379967} a^{16} + \frac{12396638543297846}{125718677761379967} a^{15} - \frac{16100363107245883}{41906225920459989} a^{14} + \frac{41672947201958432}{125718677761379967} a^{13} + \frac{3242258768931217}{41906225920459989} a^{12} + \frac{6700963571411128}{13968741973486663} a^{11} - \frac{41502094747093523}{125718677761379967} a^{10} + \frac{3735016471283644}{13968741973486663} a^{9} + \frac{5456597641494944}{125718677761379967} a^{8} - \frac{42899584155767903}{125718677761379967} a^{7} - \frac{13580886236433041}{41906225920459989} a^{6} - \frac{61958156260486238}{125718677761379967} a^{5} + \frac{53869457357751074}{125718677761379967} a^{4} - \frac{2148485652107616}{13968741973486663} a^{3} - \frac{18263293047860143}{125718677761379967} a^{2} + \frac{7042622319096799}{41906225920459989} a - \frac{20449067453149667}{41906225920459989}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1361241535}{799822089} a^{19} + \frac{1714501610}{266607363} a^{18} - \frac{1332830938}{266607363} a^{17} - \frac{7813489604}{799822089} a^{16} + \frac{689604290}{799822089} a^{15} + \frac{34892451533}{799822089} a^{14} - \frac{304076852}{88869121} a^{13} - \frac{8864849693}{88869121} a^{12} - \frac{2602753699}{88869121} a^{11} + \frac{156344842322}{799822089} a^{10} + \frac{101255242025}{799822089} a^{9} - \frac{77114168749}{266607363} a^{8} - \frac{229775571505}{799822089} a^{7} + \frac{204561280223}{799822089} a^{6} + \frac{280290281824}{799822089} a^{5} + \frac{22687365374}{799822089} a^{4} - \frac{209477807344}{799822089} a^{3} - \frac{25857866138}{266607363} a^{2} + \frac{7414654822}{266607363} a - \frac{1733409561}{88869121} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 9404.29806526 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T174:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 960
The 35 conjugacy class representatives for t20n174
Character table for t20n174 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.873.1, 5.1.1649.1, 10.0.660765843.1

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 $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ $20$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.12.0.1$x^{12} + 3 x^{2} - 2 x + 5$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$