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Properties

Label 20.0.18291664103...5625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{8}\cdot 5^{10}\cdot 7^{2}\cdot 17^{12}$
Root discriminant $23.07$
Ramified primes $3, 5, 7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T196

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -21, -16, 41, 42, -44, 63, -151, -27, 131, 89, -159, 41, 25, -2, 4, 3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 3*x^18 + 4*x^17 - 2*x^16 + 25*x^15 + 41*x^14 - 159*x^13 + 89*x^12 + 131*x^11 - 27*x^10 - 151*x^9 + 63*x^8 - 44*x^7 + 42*x^6 + 41*x^5 - 16*x^4 - 21*x^3 + 2*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 + 3*x^18 + 4*x^17 - 2*x^16 + 25*x^15 + 41*x^14 - 159*x^13 + 89*x^12 + 131*x^11 - 27*x^10 - 151*x^9 + 63*x^8 - 44*x^7 + 42*x^6 + 41*x^5 - 16*x^4 - 21*x^3 + 2*x^2 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 3 x^{18} + 4 x^{17} - 2 x^{16} + 25 x^{15} + 41 x^{14} - 159 x^{13} + 89 x^{12} + 131 x^{11} - 27 x^{10} - 151 x^{9} + 63 x^{8} - 44 x^{7} + 42 x^{6} + 41 x^{5} - 16 x^{4} - 21 x^{3} + 2 x^{2} + 5 x + 1 \)

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:  \(1829166410398033536416015625=3^{8}\cdot 5^{10}\cdot 7^{2}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.07$
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:  $3, 5, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{9} + \frac{1}{3} a$, $\frac{1}{45} a^{18} - \frac{1}{45} a^{17} + \frac{1}{9} a^{16} - \frac{1}{9} a^{15} - \frac{7}{45} a^{14} + \frac{7}{15} a^{13} + \frac{2}{9} a^{12} + \frac{1}{9} a^{11} - \frac{11}{45} a^{10} - \frac{2}{15} a^{9} + \frac{22}{45} a^{8} + \frac{19}{45} a^{7} - \frac{2}{15} a^{6} + \frac{4}{9} a^{5} - \frac{2}{45} a^{4} - \frac{1}{15} a^{3} + \frac{1}{3} a^{2} - \frac{2}{5} a - \frac{4}{45}$, $\frac{1}{454348282630474845} a^{19} + \frac{4231662434708464}{454348282630474845} a^{18} + \frac{474278856620953}{10096628502899441} a^{17} + \frac{11150706562519543}{90869656526094969} a^{16} - \frac{45962634724194137}{454348282630474845} a^{15} - \frac{9390739857830324}{454348282630474845} a^{14} + \frac{7098066320232368}{90869656526094969} a^{13} - \frac{16382226035778532}{90869656526094969} a^{12} - \frac{60554583111018526}{454348282630474845} a^{11} - \frac{210051508961778181}{454348282630474845} a^{10} + \frac{63442660622098357}{454348282630474845} a^{9} + \frac{71655439440910163}{151449427543491615} a^{8} + \frac{25025064445150139}{454348282630474845} a^{7} - \frac{40644607253781626}{90869656526094969} a^{6} - \frac{103397809453307857}{454348282630474845} a^{5} + \frac{33234425675151227}{454348282630474845} a^{4} - \frac{6544801613168252}{30289885508698323} a^{3} - \frac{11960596624827757}{50483142514497205} a^{2} - \frac{25708518584000209}{454348282630474845} a - \frac{10481956545737293}{90869656526094969}$

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:  \( 301179.447674 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T196:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1280
The 44 conjugacy class representatives for t20n196
Character table for t20n196 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.44217.1, 10.0.8553751014375.1, 10.2.6109822153125.1, 10.0.68430008115.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{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}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.0.1$x^{8} - x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
7.8.0.1$x^{8} - x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$