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Properties

Label 20.0.41316635395...1037.1
Degree $20$
Signature $[0, 10]$
Discriminant $13^{13}\cdot 97^{2}\cdot 347^{4}$
Root discriminant $26.97$
Ramified primes $13, 97, 347$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T802

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1889, -7672, 18469, -33325, 46826, -50733, 51167, -41818, 28783, -20559, 13310, -7065, 3720, -2083, 1041, -447, 183, -65, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 21*x^18 - 65*x^17 + 183*x^16 - 447*x^15 + 1041*x^14 - 2083*x^13 + 3720*x^12 - 7065*x^11 + 13310*x^10 - 20559*x^9 + 28783*x^8 - 41818*x^7 + 51167*x^6 - 50733*x^5 + 46826*x^4 - 33325*x^3 + 18469*x^2 - 7672*x + 1889)
 
gp: K = bnfinit(x^20 - 6*x^19 + 21*x^18 - 65*x^17 + 183*x^16 - 447*x^15 + 1041*x^14 - 2083*x^13 + 3720*x^12 - 7065*x^11 + 13310*x^10 - 20559*x^9 + 28783*x^8 - 41818*x^7 + 51167*x^6 - 50733*x^5 + 46826*x^4 - 33325*x^3 + 18469*x^2 - 7672*x + 1889, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 21 x^{18} - 65 x^{17} + 183 x^{16} - 447 x^{15} + 1041 x^{14} - 2083 x^{13} + 3720 x^{12} - 7065 x^{11} + 13310 x^{10} - 20559 x^{9} + 28783 x^{8} - 41818 x^{7} + 51167 x^{6} - 50733 x^{5} + 46826 x^{4} - 33325 x^{3} + 18469 x^{2} - 7672 x + 1889 \)

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:  \(41316635395822879565166861037=13^{13}\cdot 97^{2}\cdot 347^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.97$
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:  $13, 97, 347$
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}{27209283702055266887960943611500277365430851} a^{19} - \frac{8292613962335761140031879116798559577373956}{27209283702055266887960943611500277365430851} a^{18} + \frac{9364262386249455923883098499191155901471940}{27209283702055266887960943611500277365430851} a^{17} + \frac{2703782603938129032919956379424016043075904}{27209283702055266887960943611500277365430851} a^{16} + \frac{12985280426716205884442270489865656167559992}{27209283702055266887960943611500277365430851} a^{15} + \frac{10093199698389315538964750744793197234643723}{27209283702055266887960943611500277365430851} a^{14} - \frac{10798633249666314682244645184484692159190727}{27209283702055266887960943611500277365430851} a^{13} + \frac{12638208389419131680566327505561234116092764}{27209283702055266887960943611500277365430851} a^{12} - \frac{13290912862312670430170222846775589342648603}{27209283702055266887960943611500277365430851} a^{11} + \frac{8987046499493101551315035091239114079508265}{27209283702055266887960943611500277365430851} a^{10} + \frac{7516148258371332844375352380083706587640882}{27209283702055266887960943611500277365430851} a^{9} - \frac{8774924458596220193580828393985428269691278}{27209283702055266887960943611500277365430851} a^{8} + \frac{11459356925707661741930595956372594309783652}{27209283702055266887960943611500277365430851} a^{7} + \frac{1784373020682498835126601980855606667075887}{27209283702055266887960943611500277365430851} a^{6} + \frac{5184245984269043303775269034281390897861756}{27209283702055266887960943611500277365430851} a^{5} - \frac{4367081105961996564992012919887076621190388}{27209283702055266887960943611500277365430851} a^{4} - \frac{2928546303985008107952762602718964501902233}{27209283702055266887960943611500277365430851} a^{3} + \frac{7486930044608159849146648867433338845330708}{27209283702055266887960943611500277365430851} a^{2} - \frac{5994563632245321081533860420292722465815862}{27209283702055266887960943611500277365430851} a + \frac{12618207052025645535031723217141818783561111}{27209283702055266887960943611500277365430851}$

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:  $9$
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:  \( 409198.297569 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T802:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n802 are not computed
Character table for t20n802 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.3.4511.1, 10.6.44707018837.1

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 $20$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
347Data not computed