↑

Properties

Label 20.0.15509215732...8928.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 3^{16}$
Root discriminant $16.20$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
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![32, 0, 160, 0, 292, 0, 256, 0, 132, 0, 64, 0, 45, 0, 28, 0, 14, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 4*x^18 + 14*x^16 + 28*x^14 + 45*x^12 + 64*x^10 + 132*x^8 + 256*x^6 + 292*x^4 + 160*x^2 + 32)
 
gp: K = bnfinit(x^20 + 4*x^18 + 14*x^16 + 28*x^14 + 45*x^12 + 64*x^10 + 132*x^8 + 256*x^6 + 292*x^4 + 160*x^2 + 32, 1)
 

Normalized defining polynomial

\( x^{20} + 4 x^{18} + 14 x^{16} + 28 x^{14} + 45 x^{12} + 64 x^{10} + 132 x^{8} + 256 x^{6} + 292 x^{4} + 160 x^{2} + 32 \)

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:  \(1550921573240973639548928=2^{55}\cdot 3^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.20$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{9} + \frac{1}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{12} + \frac{1}{8} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{13} + \frac{1}{8} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{12} - \frac{1}{16} a^{10} + \frac{1}{16} a^{8} - \frac{1}{2} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{13} - \frac{1}{16} a^{11} + \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{32} a^{16} - \frac{1}{8} a^{10} - \frac{7}{32} a^{8} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{7}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{64} a^{17} - \frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{11}{64} a^{9} - \frac{1}{4} a^{8} - \frac{3}{16} a^{7} - \frac{1}{2} a^{6} - \frac{11}{32} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{12736} a^{18} - \frac{43}{3184} a^{16} - \frac{45}{1592} a^{14} + \frac{63}{1592} a^{12} + \frac{1289}{12736} a^{10} - \frac{1}{4} a^{9} + \frac{15}{3184} a^{8} - \frac{1}{2} a^{7} + \frac{1353}{6368} a^{6} - \frac{1}{4} a^{5} - \frac{149}{398} a^{4} + \frac{179}{796} a^{2} - \frac{13}{199}$, $\frac{1}{12736} a^{19} + \frac{27}{12736} a^{17} - \frac{45}{1592} a^{15} - \frac{73}{3184} a^{13} + \frac{493}{12736} a^{11} - \frac{2129}{12736} a^{9} - \frac{1}{4} a^{8} + \frac{159}{6368} a^{7} - \frac{1}{2} a^{6} + \frac{1795}{6368} a^{5} + \frac{1}{4} a^{4} - \frac{219}{796} a^{3} - \frac{1}{2} a^{2} + \frac{147}{796} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 104319.684091 \)
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

\(\Q(\sqrt{2}) \), 4.0.2048.2, 5.1.165888.1 x5, 10.2.220150628352.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.165888.1
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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$2$2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
3Data not computed