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Properties

Label 20.0.19883967612...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{23}\cdot 7^{10}$
Root discriminant $29.17$
Ramified primes $3, 5, 7$
Class number $4$ (GRH)
Class group $[2, 2]$ (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![1, -5, 10, -10, 30, -141, 340, -415, 220, 20, -99, 20, 220, -415, 340, -141, 30, -10, 10, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 10*x^18 - 10*x^17 + 30*x^16 - 141*x^15 + 340*x^14 - 415*x^13 + 220*x^12 + 20*x^11 - 99*x^10 + 20*x^9 + 220*x^8 - 415*x^7 + 340*x^6 - 141*x^5 + 30*x^4 - 10*x^3 + 10*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 10*x^18 - 10*x^17 + 30*x^16 - 141*x^15 + 340*x^14 - 415*x^13 + 220*x^12 + 20*x^11 - 99*x^10 + 20*x^9 + 220*x^8 - 415*x^7 + 340*x^6 - 141*x^5 + 30*x^4 - 10*x^3 + 10*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 10 x^{18} - 10 x^{17} + 30 x^{16} - 141 x^{15} + 340 x^{14} - 415 x^{13} + 220 x^{12} + 20 x^{11} - 99 x^{10} + 20 x^{9} + 220 x^{8} - 415 x^{7} + 340 x^{6} - 141 x^{5} + 30 x^{4} - 10 x^{3} + 10 x^{2} - 5 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(198839676120293140411376953125=3^{10}\cdot 5^{23}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.17$
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:  $3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{42} a^{14} - \frac{1}{42} a^{13} + \frac{1}{7} a^{12} - \frac{17}{42} a^{11} - \frac{1}{6} a^{10} + \frac{1}{7} a^{9} - \frac{17}{42} a^{8} + \frac{11}{42} a^{7} + \frac{2}{21} a^{6} - \frac{5}{14} a^{5} - \frac{1}{6} a^{4} + \frac{2}{21} a^{3} - \frac{5}{14} a^{2} - \frac{1}{42} a - \frac{10}{21}$, $\frac{1}{42} a^{15} + \frac{5}{42} a^{13} + \frac{5}{21} a^{12} + \frac{3}{7} a^{11} - \frac{1}{42} a^{10} + \frac{5}{21} a^{9} - \frac{1}{7} a^{8} + \frac{5}{14} a^{7} + \frac{5}{21} a^{6} + \frac{10}{21} a^{5} - \frac{1}{14} a^{4} + \frac{5}{21} a^{3} - \frac{8}{21} a^{2} - \frac{1}{2} a + \frac{1}{42}$, $\frac{1}{42} a^{16} - \frac{1}{7} a^{13} + \frac{3}{14} a^{12} - \frac{3}{7} a^{10} - \frac{5}{14} a^{9} + \frac{8}{21} a^{8} + \frac{3}{7} a^{7} - \frac{1}{2} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{5}{14} a^{3} + \frac{2}{7} a^{2} - \frac{5}{14} a - \frac{5}{42}$, $\frac{1}{42} a^{17} + \frac{1}{14} a^{13} - \frac{1}{7} a^{12} + \frac{1}{7} a^{11} - \frac{5}{14} a^{10} + \frac{5}{21} a^{9} + \frac{1}{14} a^{7} + \frac{2}{7} a^{6} + \frac{3}{7} a^{5} - \frac{5}{14} a^{4} - \frac{1}{7} a^{3} - \frac{1}{2} a^{2} - \frac{11}{42} a + \frac{1}{7}$, $\frac{1}{48342} a^{18} - \frac{7}{1151} a^{17} - \frac{199}{48342} a^{16} + \frac{35}{6906} a^{15} - \frac{365}{48342} a^{14} - \frac{324}{8057} a^{13} + \frac{7739}{48342} a^{12} + \frac{8257}{48342} a^{11} + \frac{8201}{24171} a^{10} + \frac{3047}{6906} a^{9} - \frac{3357}{16114} a^{8} + \frac{7006}{24171} a^{7} - \frac{1257}{16114} a^{6} + \frac{17623}{48342} a^{5} - \frac{4211}{24171} a^{4} + \frac{1000}{8057} a^{3} - \frac{20917}{48342} a^{2} - \frac{6049}{48342} a - \frac{7481}{24171}$, $\frac{1}{48342} a^{19} - \frac{155}{24171} a^{17} + \frac{220}{24171} a^{16} + \frac{101}{16114} a^{15} + \frac{13}{6906} a^{14} - \frac{11317}{48342} a^{13} - \frac{551}{2302} a^{12} + \frac{11897}{48342} a^{11} + \frac{1331}{6906} a^{10} + \frac{22235}{48342} a^{9} - \frac{2019}{16114} a^{8} + \frac{4385}{48342} a^{7} + \frac{7003}{48342} a^{6} + \frac{281}{2302} a^{5} - \frac{3464}{24171} a^{4} - \frac{4135}{48342} a^{3} + \frac{6873}{16114} a^{2} - \frac{10469}{48342} a + \frac{3502}{8057}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 1126766.82054 \) (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

\(\Q(\sqrt{5}) \), 4.0.55125.1, 5.1.1378125.1 x5, 10.2.9496142578125.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.1378125.1
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5Data not computed
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$