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Properties

Label 20.8.15583578925...2688.4
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 3^{5}\cdot 11^{19}$
Root discriminant $25.68$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T427

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![67, -546, 2080, -5226, 10123, -15956, 20369, -20492, 15328, -6804, -1353, 6048, -6674, 4750, -2330, 698, -32, -82, 42, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 42*x^18 - 82*x^17 - 32*x^16 + 698*x^15 - 2330*x^14 + 4750*x^13 - 6674*x^12 + 6048*x^11 - 1353*x^10 - 6804*x^9 + 15328*x^8 - 20492*x^7 + 20369*x^6 - 15956*x^5 + 10123*x^4 - 5226*x^3 + 2080*x^2 - 546*x + 67)
 
gp: K = bnfinit(x^20 - 10*x^19 + 42*x^18 - 82*x^17 - 32*x^16 + 698*x^15 - 2330*x^14 + 4750*x^13 - 6674*x^12 + 6048*x^11 - 1353*x^10 - 6804*x^9 + 15328*x^8 - 20492*x^7 + 20369*x^6 - 15956*x^5 + 10123*x^4 - 5226*x^3 + 2080*x^2 - 546*x + 67, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 42 x^{18} - 82 x^{17} - 32 x^{16} + 698 x^{15} - 2330 x^{14} + 4750 x^{13} - 6674 x^{12} + 6048 x^{11} - 1353 x^{10} - 6804 x^{9} + 15328 x^{8} - 20492 x^{7} + 20369 x^{6} - 15956 x^{5} + 10123 x^{4} - 5226 x^{3} + 2080 x^{2} - 546 x + 67 \)

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:  \(15583578925526925703866482688=2^{20}\cdot 3^{5}\cdot 11^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.68$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{73933235597949994189397} a^{19} + \frac{18151795787106656411617}{73933235597949994189397} a^{18} + \frac{26514583220869235784301}{73933235597949994189397} a^{17} - \frac{13747948044513580796373}{73933235597949994189397} a^{16} + \frac{43113157855238388533}{73933235597949994189397} a^{15} - \frac{18175632370917183307523}{73933235597949994189397} a^{14} - \frac{24319338448435882493109}{73933235597949994189397} a^{13} + \frac{12633420305147644809211}{73933235597949994189397} a^{12} + \frac{26598820015972591005760}{73933235597949994189397} a^{11} + \frac{15544175390173517798458}{73933235597949994189397} a^{10} - \frac{30000600686577574121803}{73933235597949994189397} a^{9} - \frac{34535468257745621515219}{73933235597949994189397} a^{8} - \frac{12569158017156576464719}{73933235597949994189397} a^{7} - \frac{2192322385076071404174}{73933235597949994189397} a^{6} + \frac{36254719073961567953263}{73933235597949994189397} a^{5} - \frac{17453572321425246683335}{73933235597949994189397} a^{4} - \frac{29361386511970096346493}{73933235597949994189397} a^{3} + \frac{17791044731006651666476}{73933235597949994189397} a^{2} - \frac{1352718799341804082897}{3214488504258695399539} a - \frac{16416370156401834321959}{73933235597949994189397}$

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:  \( 3406311.30931 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T427:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n427 are not computed
Character table for t20n427 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.2414538435584.1

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 R R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ $20$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
2Data not computed
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.0.1$x^{10} - x^{3} - x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11Data not computed