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Properties

Label 20.10.9228883441...1648.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{20}\cdot 11^{18}\cdot 1583$
Root discriminant $25.02$
Ramified primes $2, 11, 1583$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T409

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -7, 54, -88, 56, -48, 136, -210, 154, -87, 154, -210, 136, -48, 56, -88, 54, -7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 7*x^18 + 54*x^17 - 88*x^16 + 56*x^15 - 48*x^14 + 136*x^13 - 210*x^12 + 154*x^11 - 87*x^10 + 154*x^9 - 210*x^8 + 136*x^7 - 48*x^6 + 56*x^5 - 88*x^4 + 54*x^3 - 7*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 - 7*x^18 + 54*x^17 - 88*x^16 + 56*x^15 - 48*x^14 + 136*x^13 - 210*x^12 + 154*x^11 - 87*x^10 + 154*x^9 - 210*x^8 + 136*x^7 - 48*x^6 + 56*x^5 - 88*x^4 + 54*x^3 - 7*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 7 x^{18} + 54 x^{17} - 88 x^{16} + 56 x^{15} - 48 x^{14} + 136 x^{13} - 210 x^{12} + 154 x^{11} - 87 x^{10} + 154 x^{9} - 210 x^{8} + 136 x^{7} - 48 x^{6} + 56 x^{5} - 88 x^{4} + 54 x^{3} - 7 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-9228883441492376875877531648=-\,2^{20}\cdot 11^{18}\cdot 1583\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.02$
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:  $2, 11, 1583$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{89} a^{16} - \frac{23}{89} a^{15} + \frac{40}{89} a^{14} - \frac{32}{89} a^{13} + \frac{41}{89} a^{12} - \frac{16}{89} a^{11} - \frac{36}{89} a^{10} - \frac{6}{89} a^{9} - \frac{33}{89} a^{8} - \frac{6}{89} a^{7} - \frac{36}{89} a^{6} - \frac{16}{89} a^{5} + \frac{41}{89} a^{4} - \frac{32}{89} a^{3} + \frac{40}{89} a^{2} - \frac{23}{89} a + \frac{1}{89}$, $\frac{1}{89} a^{17} - \frac{44}{89} a^{15} - \frac{2}{89} a^{14} + \frac{17}{89} a^{13} + \frac{37}{89} a^{12} + \frac{41}{89} a^{11} - \frac{33}{89} a^{10} + \frac{7}{89} a^{9} + \frac{36}{89} a^{8} + \frac{4}{89} a^{7} - \frac{43}{89} a^{6} + \frac{29}{89} a^{5} + \frac{21}{89} a^{4} + \frac{16}{89} a^{3} + \frac{7}{89} a^{2} + \frac{6}{89} a + \frac{23}{89}$, $\frac{1}{3827} a^{18} - \frac{19}{3827} a^{17} + \frac{19}{3827} a^{16} - \frac{1416}{3827} a^{15} - \frac{1786}{3827} a^{14} + \frac{1258}{3827} a^{13} - \frac{749}{3827} a^{12} + \frac{1384}{3827} a^{11} + \frac{591}{3827} a^{10} - \frac{119}{3827} a^{9} - \frac{18}{43} a^{8} + \frac{1728}{3827} a^{7} - \frac{1867}{3827} a^{6} + \frac{1043}{3827} a^{5} + \frac{1310}{3827} a^{4} + \frac{1336}{3827} a^{3} - \frac{1701}{3827} a^{2} + \frac{1486}{3827} a - \frac{730}{3827}$, $\frac{1}{3827} a^{19} + \frac{2}{3827} a^{17} + \frac{20}{3827} a^{16} + \frac{335}{3827} a^{15} - \frac{1845}{3827} a^{14} - \frac{1572}{3827} a^{13} + \frac{1859}{3827} a^{12} + \frac{829}{3827} a^{11} - \frac{672}{3827} a^{10} - \frac{251}{3827} a^{9} + \frac{1777}{3827} a^{8} - \frac{898}{3827} a^{7} + \frac{99}{3827} a^{6} - \frac{1405}{3827} a^{5} + \frac{985}{3827} a^{4} - \frac{1386}{3827} a^{3} - \frac{733}{3827} a^{2} + \frac{1016}{3827} a - \frac{1056}{3827}$

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:  $14$
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:  \( 3944561.84803 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T409:

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 t20n409 are not computed
Character table for t20n409 is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\)

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{14}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
1583Data not computed