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Properties

Label 20.12.1541817086...0000.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{30}\cdot 5^{5}\cdot 11^{16}$
Root discriminant $28.80$
Ramified primes $2, 5, 11$
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, 22, 1192, -742, -6956, 5846, 14254, -13214, -9163, 8016, 5104, -3230, -2779, 1436, 754, -454, -55, 70, -8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 8*x^18 + 70*x^17 - 55*x^16 - 454*x^15 + 754*x^14 + 1436*x^13 - 2779*x^12 - 3230*x^11 + 5104*x^10 + 8016*x^9 - 9163*x^8 - 13214*x^7 + 14254*x^6 + 5846*x^5 - 6956*x^4 - 742*x^3 + 1192*x^2 + 22*x - 67)
 
gp: K = bnfinit(x^20 - 4*x^19 - 8*x^18 + 70*x^17 - 55*x^16 - 454*x^15 + 754*x^14 + 1436*x^13 - 2779*x^12 - 3230*x^11 + 5104*x^10 + 8016*x^9 - 9163*x^8 - 13214*x^7 + 14254*x^6 + 5846*x^5 - 6956*x^4 - 742*x^3 + 1192*x^2 + 22*x - 67, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 8 x^{18} + 70 x^{17} - 55 x^{16} - 454 x^{15} + 754 x^{14} + 1436 x^{13} - 2779 x^{12} - 3230 x^{11} + 5104 x^{10} + 8016 x^{9} - 9163 x^{8} - 13214 x^{7} + 14254 x^{6} + 5846 x^{5} - 6956 x^{4} - 742 x^{3} + 1192 x^{2} + 22 x - 67 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(154181708612560135336755200000=2^{30}\cdot 5^{5}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.80$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{5520119933881251665205392683634501} a^{19} + \frac{897748266440298877051301205873743}{5520119933881251665205392683634501} a^{18} + \frac{1840107504324840710429643404457251}{5520119933881251665205392683634501} a^{17} - \frac{91616551081685877895252049403466}{5520119933881251665205392683634501} a^{16} - \frac{984693693261114830371495198748885}{5520119933881251665205392683634501} a^{15} - \frac{855272758902223759389881098177583}{5520119933881251665205392683634501} a^{14} + \frac{1543473608330347794961894014492656}{5520119933881251665205392683634501} a^{13} - \frac{887777426894300129823356660614961}{5520119933881251665205392683634501} a^{12} - \frac{114064067145349142166315725153373}{5520119933881251665205392683634501} a^{11} + \frac{335291739128579608058664822445466}{5520119933881251665205392683634501} a^{10} + \frac{544264691939415408595252295470923}{5520119933881251665205392683634501} a^{9} - \frac{2358525599601525237883285858493461}{5520119933881251665205392683634501} a^{8} + \frac{731355223275415195053547850150028}{5520119933881251665205392683634501} a^{7} - \frac{1521041134337918113605174276399177}{5520119933881251665205392683634501} a^{6} + \frac{808276465315285479870043080930875}{5520119933881251665205392683634501} a^{5} - \frac{1994370049588424010483532135463267}{5520119933881251665205392683634501} a^{4} - \frac{1607251792358647254012477608907971}{5520119933881251665205392683634501} a^{3} + \frac{1718133009947145169786990587612985}{5520119933881251665205392683634501} a^{2} - \frac{1946726609222119719017593018967475}{5520119933881251665205392683634501} a - \frac{806704937047609162873464158988692}{5520119933881251665205392683634501}$

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:  $15$
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:  \( 21332568.9868 \) (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

\(\Q(\zeta_{11})^+\), 10.8.219503494144.1

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 $20$ R $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$5$5.10.0.1$x^{10} + x^{2} - x + 3$$1$$10$$0$$C_{10}$$[\ ]^{10}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$