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Properties

Label 20.0.15233950180...4384.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{10}\cdot 89^{8}$
Root discriminant $18.16$
Ramified primes $2, 3, 89$
Class number $2$
Class group $[2]$
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![16, 16, 32, 16, 32, 8, 24, 8, 48, -16, 48, 8, 32, -12, 32, -14, 16, -8, 6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 6*x^18 - 8*x^17 + 16*x^16 - 14*x^15 + 32*x^14 - 12*x^13 + 32*x^12 + 8*x^11 + 48*x^10 - 16*x^9 + 48*x^8 + 8*x^7 + 24*x^6 + 8*x^5 + 32*x^4 + 16*x^3 + 32*x^2 + 16*x + 16)
 
gp: K = bnfinit(x^20 - 2*x^19 + 6*x^18 - 8*x^17 + 16*x^16 - 14*x^15 + 32*x^14 - 12*x^13 + 32*x^12 + 8*x^11 + 48*x^10 - 16*x^9 + 48*x^8 + 8*x^7 + 24*x^6 + 8*x^5 + 32*x^4 + 16*x^3 + 32*x^2 + 16*x + 16, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 6 x^{18} - 8 x^{17} + 16 x^{16} - 14 x^{15} + 32 x^{14} - 12 x^{13} + 32 x^{12} + 8 x^{11} + 48 x^{10} - 16 x^{9} + 48 x^{8} + 8 x^{7} + 24 x^{6} + 8 x^{5} + 32 x^{4} + 16 x^{3} + 32 x^{2} + 16 x + 16 \)

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:  \(15233950180173557331984384=2^{16}\cdot 3^{10}\cdot 89^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.16$
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:  $2, 3, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{4} a^{13}$, $\frac{1}{12} a^{14} + \frac{1}{12} a^{11} + \frac{1}{12} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{24} a^{15} - \frac{1}{12} a^{12} - \frac{1}{12} a^{11} - \frac{1}{12} a^{10} + \frac{1}{6} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{24} a^{16} - \frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{1}{12} a^{11} + \frac{1}{6} a^{9} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{24} a^{17} - \frac{1}{12} a^{13} - \frac{1}{12} a^{12} + \frac{1}{12} a^{11} - \frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{2136} a^{18} - \frac{13}{712} a^{17} - \frac{7}{534} a^{16} - \frac{5}{534} a^{15} - \frac{37}{1068} a^{14} - \frac{11}{178} a^{13} + \frac{29}{534} a^{12} - \frac{13}{178} a^{11} + \frac{1}{178} a^{10} + \frac{1}{6} a^{9} + \frac{20}{89} a^{8} + \frac{95}{534} a^{7} - \frac{73}{534} a^{6} - \frac{3}{89} a^{5} + \frac{1}{267} a^{4} - \frac{4}{267} a^{3} + \frac{10}{267} a^{2} + \frac{37}{89} a + \frac{28}{89}$, $\frac{1}{98256} a^{19} - \frac{1}{24564} a^{18} + \frac{475}{24564} a^{17} + \frac{53}{12282} a^{16} - \frac{209}{49128} a^{15} - \frac{13}{24564} a^{14} + \frac{663}{8188} a^{13} - \frac{1381}{12282} a^{12} + \frac{863}{24564} a^{11} - \frac{793}{12282} a^{10} - \frac{860}{6141} a^{9} + \frac{1391}{12282} a^{8} - \frac{77}{6141} a^{7} + \frac{1873}{12282} a^{6} - \frac{937}{12282} a^{5} + \frac{5371}{12282} a^{4} - \frac{332}{6141} a^{3} + \frac{329}{2047} a^{2} - \frac{2777}{6141} a - \frac{2357}{6141}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{2533}{98256} a^{19} + \frac{1153}{24564} a^{18} - \frac{3035}{24564} a^{17} + \frac{875}{6141} a^{16} - \frac{13127}{49128} a^{15} + \frac{1347}{8188} a^{14} - \frac{12775}{24564} a^{13} + \frac{1133}{24564} a^{12} - \frac{9673}{24564} a^{11} - \frac{1577}{12282} a^{10} - \frac{7816}{6141} a^{9} + \frac{2030}{2047} a^{8} - \frac{13361}{12282} a^{7} - \frac{268}{6141} a^{6} - \frac{1797}{2047} a^{5} - \frac{1807}{12282} a^{4} - \frac{1899}{2047} a^{3} + \frac{374}{6141} a^{2} - \frac{2283}{2047} a + \frac{2747}{6141} \) (order $6$)
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:  \( 64033.5089322 \)
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

\(\Q(\sqrt{-3}) \), 10.4.433674369792.1

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

Sibling fields

Degree 5 sibling: 5.3.380208.1
Degree 6 sibling: 6.0.3421872.4
Degree 10 siblings: 10.0.3903069328128.1, 10.4.433674369792.1
Degree 12 sibling: 12.0.11709207984384.1
Degree 15 sibling: 15.3.1483978183108890624.1
Degree 20 siblings: Deg 20, 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$