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Properties

Label 20.0.13115313200...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 73^{10}$
Root discriminant $28.57$
Ramified primes $5, 73$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -125, 415, -1100, 2436, -4061, 5114, -5271, 5490, -5361, 3747, -23, -2082, 1337, 137, -402, 94, 48, -17, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 17*x^18 + 48*x^17 + 94*x^16 - 402*x^15 + 137*x^14 + 1337*x^13 - 2082*x^12 - 23*x^11 + 3747*x^10 - 5361*x^9 + 5490*x^8 - 5271*x^7 + 5114*x^6 - 4061*x^5 + 2436*x^4 - 1100*x^3 + 415*x^2 - 125*x + 25)
 
gp: K = bnfinit(x^20 - 2*x^19 - 17*x^18 + 48*x^17 + 94*x^16 - 402*x^15 + 137*x^14 + 1337*x^13 - 2082*x^12 - 23*x^11 + 3747*x^10 - 5361*x^9 + 5490*x^8 - 5271*x^7 + 5114*x^6 - 4061*x^5 + 2436*x^4 - 1100*x^3 + 415*x^2 - 125*x + 25, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 17 x^{18} + 48 x^{17} + 94 x^{16} - 402 x^{15} + 137 x^{14} + 1337 x^{13} - 2082 x^{12} - 23 x^{11} + 3747 x^{10} - 5361 x^{9} + 5490 x^{8} - 5271 x^{7} + 5114 x^{6} - 4061 x^{5} + 2436 x^{4} - 1100 x^{3} + 415 x^{2} - 125 x + 25 \)

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:  \(131153132009996266143798828125=5^{15}\cdot 73^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.57$
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:  $5, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{20} a^{14} - \frac{3}{20} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{5} a^{10} - \frac{1}{4} a^{9} - \frac{9}{20} a^{8} - \frac{1}{4} a^{7} + \frac{3}{10} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{10} a^{3} - \frac{1}{5} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{20} a^{15} - \frac{1}{5} a^{13} + \frac{1}{20} a^{11} + \frac{3}{20} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{20} a^{7} + \frac{3}{20} a^{6} - \frac{1}{4} a^{5} + \frac{2}{5} a^{4} - \frac{7}{20} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{20} a^{16} - \frac{1}{10} a^{13} + \frac{1}{20} a^{12} + \frac{3}{20} a^{11} - \frac{1}{10} a^{9} - \frac{1}{4} a^{8} - \frac{7}{20} a^{7} + \frac{9}{20} a^{6} - \frac{1}{10} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{5} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{20} a^{17} - \frac{1}{4} a^{13} + \frac{3}{20} a^{12} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{9}{20} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{7}{20} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{20} a^{18} - \frac{1}{10} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{5} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{7}{20} a^{3} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{3223630932786236148968200} a^{19} - \frac{53646644089040189969941}{3223630932786236148968200} a^{18} - \frac{11909521677784311290319}{1611815466393118074484100} a^{17} + \frac{1521825442458969221923}{80590773319655903724205} a^{16} - \frac{13060636935865832668353}{1611815466393118074484100} a^{15} - \frac{10907023327312681687349}{1611815466393118074484100} a^{14} - \frac{333265722440596690436581}{3223630932786236148968200} a^{13} - \frac{129468342200769916787167}{1611815466393118074484100} a^{12} + \frac{284857687577187493769647}{1611815466393118074484100} a^{11} + \frac{389621218125655679436921}{3223630932786236148968200} a^{10} - \frac{246928245003477235823251}{1611815466393118074484100} a^{9} - \frac{355851290791106106996973}{3223630932786236148968200} a^{8} - \frac{13730303681691827744073}{3223630932786236148968200} a^{7} + \frac{183627898192003545243562}{402953866598279518621025} a^{6} - \frac{71011758821067935917337}{161181546639311807448410} a^{5} - \frac{1183311056329360798404211}{3223630932786236148968200} a^{4} - \frac{7109690878070219570781}{128945237311449445958728} a^{3} + \frac{77375621945640487630143}{644726186557247229793640} a^{2} + \frac{13558521267518352644473}{64472618655724722979364} a - \frac{56375402073294721421203}{128945237311449445958728}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1811157.39147 \) (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

\(\Q(\sqrt{5}) \), 4.0.666125.1, 5.1.666125.1 x5, 10.2.2218612578125.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.666125.1
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$73$73.4.2.2$x^{4} - 73 x^{2} + 58619$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
73.4.2.2$x^{4} - 73 x^{2} + 58619$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
73.4.2.2$x^{4} - 73 x^{2} + 58619$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
73.4.2.2$x^{4} - 73 x^{2} + 58619$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
73.4.2.2$x^{4} - 73 x^{2} + 58619$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$