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Properties

Label 20.4.28888165452...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{36}\cdot 3^{16}\cdot 5^{10}$
Root discriminant $18.75$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $S_5$ (as 20T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-31, 58, 327, 396, 149, -4, 62, 164, 61, -150, -119, 28, 21, -16, 2, 8, 5, -2, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 - 2*x^17 + 5*x^16 + 8*x^15 + 2*x^14 - 16*x^13 + 21*x^12 + 28*x^11 - 119*x^10 - 150*x^9 + 61*x^8 + 164*x^7 + 62*x^6 - 4*x^5 + 149*x^4 + 396*x^3 + 327*x^2 + 58*x - 31)
 
gp: K = bnfinit(x^20 - 5*x^18 - 2*x^17 + 5*x^16 + 8*x^15 + 2*x^14 - 16*x^13 + 21*x^12 + 28*x^11 - 119*x^10 - 150*x^9 + 61*x^8 + 164*x^7 + 62*x^6 - 4*x^5 + 149*x^4 + 396*x^3 + 327*x^2 + 58*x - 31, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{18} - 2 x^{17} + 5 x^{16} + 8 x^{15} + 2 x^{14} - 16 x^{13} + 21 x^{12} + 28 x^{11} - 119 x^{10} - 150 x^{9} + 61 x^{8} + 164 x^{7} + 62 x^{6} - 4 x^{5} + 149 x^{4} + 396 x^{3} + 327 x^{2} + 58 x - 31 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(28888165452349440000000000=2^{36}\cdot 3^{16}\cdot 5^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.75$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{14} + \frac{1}{10} a^{13} - \frac{1}{2} a^{11} + \frac{1}{10} a^{10} + \frac{2}{5} a^{9} + \frac{1}{10} a^{8} + \frac{3}{10} a^{7} + \frac{3}{10} a^{6} - \frac{3}{10} a^{4} + \frac{3}{10} a^{3} - \frac{2}{5} a^{2} + \frac{1}{10} a - \frac{1}{10}$, $\frac{1}{30} a^{15} + \frac{1}{30} a^{14} - \frac{1}{6} a^{13} - \frac{7}{15} a^{11} - \frac{1}{30} a^{10} - \frac{3}{10} a^{9} + \frac{13}{30} a^{8} - \frac{2}{5} a^{7} - \frac{1}{6} a^{6} - \frac{13}{30} a^{5} - \frac{7}{30} a^{4} + \frac{11}{30} a^{3} - \frac{7}{15} a^{2} - \frac{1}{5} a + \frac{1}{6}$, $\frac{1}{30} a^{16} - \frac{2}{15} a^{13} + \frac{1}{30} a^{12} - \frac{1}{15} a^{11} + \frac{13}{30} a^{10} - \frac{7}{15} a^{9} + \frac{11}{30} a^{8} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{5} a^{5} + \frac{4}{15} a^{3} - \frac{1}{30} a^{2} + \frac{1}{15} a + \frac{2}{15}$, $\frac{1}{30} a^{17} - \frac{1}{30} a^{14} + \frac{2}{15} a^{13} - \frac{1}{15} a^{12} - \frac{1}{15} a^{11} - \frac{11}{30} a^{10} - \frac{7}{30} a^{9} + \frac{13}{30} a^{8} + \frac{2}{15} a^{7} - \frac{1}{2} a^{6} - \frac{1}{30} a^{4} + \frac{4}{15} a^{3} - \frac{1}{3} a^{2} + \frac{7}{30} a - \frac{1}{10}$, $\frac{1}{30} a^{18} - \frac{1}{30} a^{14} + \frac{1}{15} a^{13} - \frac{1}{15} a^{12} - \frac{1}{3} a^{11} - \frac{7}{15} a^{10} + \frac{1}{3} a^{9} + \frac{11}{30} a^{8} + \frac{7}{30} a^{6} - \frac{7}{15} a^{5} - \frac{11}{30} a^{4} - \frac{1}{15} a^{3} - \frac{13}{30} a^{2} + \frac{11}{30}$, $\frac{1}{16161150287147490} a^{19} - \frac{13233815721611}{3232230057429498} a^{18} + \frac{814429034923}{8080575143573745} a^{17} - \frac{284275116197}{950655899243970} a^{16} + \frac{19620830308289}{2693525047857915} a^{15} - \frac{9372384948193}{190131179848794} a^{14} - \frac{585834654182948}{2693525047857915} a^{13} - \frac{2697074332119661}{16161150287147490} a^{12} + \frac{3709842647358247}{16161150287147490} a^{11} + \frac{43497074773771}{114618087142890} a^{10} - \frac{626435324286352}{8080575143573745} a^{9} + \frac{3833432886069589}{8080575143573745} a^{8} + \frac{40556800168939}{8080575143573745} a^{7} + \frac{286985649999458}{2693525047857915} a^{6} + \frac{265406458596749}{1616115028714749} a^{5} - \frac{403204547641405}{1077410019143166} a^{4} + \frac{1598553130143121}{3232230057429498} a^{3} + \frac{81072122327407}{475327949621985} a^{2} + \frac{2270531879009167}{8080575143573745} a + \frac{2387197663651487}{16161150287147490}$

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:  $11$
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:  \( 110122.225791 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_5$ (as 20T32):

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

\(\Q(\sqrt{5}) \), 10.2.214990848000.1

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

Sibling fields

Degree 5 sibling: 5.1.103680.1
Degree 6 sibling: 6.2.10368000.1
Degree 10 siblings: 10.2.214990848000.1, 10.2.1343692800000.3
Degree 12 sibling: 12.4.107495424000000.1
Degree 15 sibling: 15.3.2229025112064000000.1
Degree 20 siblings: 20.0.369768517790072832000000000.1, Deg 20
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 R R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.8.16.9$x^{8} + 2 x^{4} + 8 x + 12$$4$$2$$16$$S_4$$[8/3, 8/3]_{3}^{2}$
2.12.20.34$x^{12} + 14 x^{10} + 16 x^{8} - 8 x^{6} - 8 x^{4} + 16 x^{2} + 16$$6$$2$$20$$S_4$$[8/3, 8/3]_{3}^{2}$
$3$3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$