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Properties

Label 20.6.11805220867...5887.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,3^{16}\cdot 223^{7}$
Root discriminant $15.98$
Ramified primes $3, 223$
Class number $1$
Class group Trivial
Galois group $S_5$ (as 20T30)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 12, 2, -8, -11, -55, 156, -25, -148, 123, 32, -82, 21, 41, -23, -5, 11, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 3*x^18 + 11*x^17 - 5*x^16 - 23*x^15 + 41*x^14 + 21*x^13 - 82*x^12 + 32*x^11 + 123*x^10 - 148*x^9 - 25*x^8 + 156*x^7 - 55*x^6 - 11*x^5 - 8*x^4 + 2*x^3 + 12*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^20 - x^19 - 3*x^18 + 11*x^17 - 5*x^16 - 23*x^15 + 41*x^14 + 21*x^13 - 82*x^12 + 32*x^11 + 123*x^10 - 148*x^9 - 25*x^8 + 156*x^7 - 55*x^6 - 11*x^5 - 8*x^4 + 2*x^3 + 12*x^2 - 7*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 3 x^{18} + 11 x^{17} - 5 x^{16} - 23 x^{15} + 41 x^{14} + 21 x^{13} - 82 x^{12} + 32 x^{11} + 123 x^{10} - 148 x^{9} - 25 x^{8} + 156 x^{7} - 55 x^{6} - 11 x^{5} - 8 x^{4} + 2 x^{3} + 12 x^{2} - 7 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1180522086783240923185887=-\,3^{16}\cdot 223^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.98$
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:  $3, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{17} - \frac{1}{2} a^{15} + \frac{1}{4} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{344} a^{18} + \frac{7}{344} a^{17} + \frac{1}{172} a^{16} + \frac{7}{172} a^{15} + \frac{5}{344} a^{14} - \frac{9}{344} a^{13} + \frac{29}{172} a^{12} - \frac{11}{43} a^{11} + \frac{5}{43} a^{10} + \frac{3}{43} a^{9} - \frac{5}{344} a^{8} - \frac{9}{86} a^{7} - \frac{29}{172} a^{6} + \frac{19}{43} a^{5} - \frac{9}{344} a^{4} + \frac{77}{344} a^{3} + \frac{35}{344} a^{2} + \frac{139}{344} a + \frac{27}{344}$, $\frac{1}{77934279610288} a^{19} - \frac{24218732933}{19483569902572} a^{18} - \frac{4366186418071}{77934279610288} a^{17} + \frac{2300592791967}{9741784951286} a^{16} - \frac{20394354319757}{77934279610288} a^{15} + \frac{2100746284028}{4870892475643} a^{14} + \frac{17940863806329}{77934279610288} a^{13} + \frac{4010189181441}{38967139805144} a^{12} - \frac{3314452513895}{9741784951286} a^{11} - \frac{4229526163223}{9741784951286} a^{10} - \frac{34996514408765}{77934279610288} a^{9} - \frac{32717854424541}{77934279610288} a^{8} + \frac{12446223807055}{38967139805144} a^{7} - \frac{9936036383387}{38967139805144} a^{6} - \frac{17058377469997}{77934279610288} a^{5} - \frac{7788406357435}{19483569902572} a^{4} - \frac{2631374881043}{19483569902572} a^{3} + \frac{6998835177163}{38967139805144} a^{2} - \frac{3523960468143}{38967139805144} a - \frac{37663738270797}{77934279610288}$

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:  $12$
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:  \( 14563.3790161 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_5$ (as 20T30):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 7 conjugacy class representatives for $S_5$
Character table for $S_5$

Intermediate fields

5.3.18063.1 x2, 10.4.72758649087.1

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

Sibling fields

Degree 5 sibling: data not computed
Degree 6 sibling: data not computed
Degree 10 siblings: data not computed
Degree 12 sibling: data not computed
Degree 15 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
3Data not computed
223Data not computed