↑

Properties

Label 20.0.11609689553...1501.1
Degree $20$
Signature $[0, 10]$
Discriminant $101^{15}$
Root discriminant $31.86$
Ramified prime $101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_5$ (as 20T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![101, 0, -1313, 0, 4714, 0, 12402, 0, 7923, 0, 308, 0, -999, 0, 13, 0, 82, 0, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 16*x^18 + 82*x^16 + 13*x^14 - 999*x^12 + 308*x^10 + 7923*x^8 + 12402*x^6 + 4714*x^4 - 1313*x^2 + 101)
 
gp: K = bnfinit(x^20 - 16*x^18 + 82*x^16 + 13*x^14 - 999*x^12 + 308*x^10 + 7923*x^8 + 12402*x^6 + 4714*x^4 - 1313*x^2 + 101, 1)
 

Normalized defining polynomial

\( x^{20} - 16 x^{18} + 82 x^{16} + 13 x^{14} - 999 x^{12} + 308 x^{10} + 7923 x^{8} + 12402 x^{6} + 4714 x^{4} - 1313 x^{2} + 101 \)

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:  \(1160968955369998535166956051501=101^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.86$
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:  $101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{38} a^{12} + \frac{4}{19} a^{10} - \frac{1}{38} a^{8} + \frac{5}{38} a^{6} - \frac{1}{2} a^{5} + \frac{9}{19} a^{4} + \frac{4}{19} a^{2} - \frac{9}{19}$, $\frac{1}{38} a^{13} + \frac{4}{19} a^{11} - \frac{1}{38} a^{9} + \frac{5}{38} a^{7} - \frac{1}{2} a^{6} + \frac{9}{19} a^{5} + \frac{4}{19} a^{3} - \frac{9}{19} a$, $\frac{1}{76} a^{14} - \frac{2}{19} a^{10} - \frac{3}{38} a^{8} + \frac{35}{76} a^{6} + \frac{4}{19} a^{4} - \frac{1}{2} a^{3} + \frac{13}{76} a^{2} + \frac{11}{76}$, $\frac{1}{76} a^{15} - \frac{2}{19} a^{11} - \frac{3}{38} a^{9} - \frac{3}{76} a^{7} + \frac{4}{19} a^{5} - \frac{1}{2} a^{4} - \frac{25}{76} a^{3} - \frac{27}{76} a - \frac{1}{2}$, $\frac{1}{76} a^{16} - \frac{9}{38} a^{10} - \frac{11}{76} a^{8} - \frac{5}{19} a^{6} - \frac{1}{2} a^{5} - \frac{33}{76} a^{4} + \frac{37}{76} a^{2} - \frac{1}{2} a + \frac{2}{19}$, $\frac{1}{152} a^{17} - \frac{1}{152} a^{14} - \frac{1}{76} a^{13} - \frac{17}{76} a^{11} - \frac{15}{76} a^{10} + \frac{29}{152} a^{9} + \frac{3}{76} a^{8} - \frac{15}{76} a^{7} - \frac{73}{152} a^{6} - \frac{31}{152} a^{5} - \frac{27}{76} a^{4} - \frac{17}{152} a^{3} - \frac{13}{152} a^{2} + \frac{11}{38} a - \frac{11}{152}$, $\frac{1}{1471888229032} a^{18} - \frac{1427556399}{735944114516} a^{16} - \frac{1}{152} a^{15} - \frac{226840687}{183986028629} a^{14} - \frac{9273179189}{735944114516} a^{12} - \frac{15}{76} a^{11} + \frac{17202381553}{1471888229032} a^{10} + \frac{3}{76} a^{9} + \frac{29719545997}{183986028629} a^{8} + \frac{3}{152} a^{7} + \frac{698008082927}{1471888229032} a^{6} + \frac{11}{76} a^{5} - \frac{673971733355}{1471888229032} a^{4} - \frac{13}{152} a^{3} - \frac{83759586670}{183986028629} a^{2} - \frac{11}{152} a + \frac{205172358101}{735944114516}$, $\frac{1}{1471888229032} a^{19} - \frac{1427556399}{735944114516} a^{17} - \frac{1}{152} a^{16} - \frac{226840687}{183986028629} a^{15} - \frac{9273179189}{735944114516} a^{13} - \frac{1}{76} a^{12} + \frac{17202381553}{1471888229032} a^{11} + \frac{1}{76} a^{10} + \frac{29719545997}{183986028629} a^{9} - \frac{25}{152} a^{8} - \frac{37936031589}{1471888229032} a^{7} - \frac{33}{76} a^{6} + \frac{61972381161}{1471888229032} a^{5} - \frac{41}{152} a^{4} - \frac{83759586670}{183986028629} a^{3} + \frac{61}{152} a^{2} - \frac{162799699157}{735944114516} a + \frac{7}{38}$

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:  $9$
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:  \( 11240237.3489 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{101}) \), 4.0.1030301.1, 5.1.1030301.1 x5, 10.2.107213535210701.3 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 5 sibling: 5.1.1030301.1
Degree 10 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$101$101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$