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Properties

Label 20.20.8501150111...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{36}\cdot 5^{10}\cdot 103^{8}$
Root discriminant $49.71$
Ramified primes $2, 5, 103$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![20, 600, 5396, 19672, 22258, -33928, -86922, -964, 101057, 31228, -57969, -23720, 19008, 8012, -3799, -1392, 468, 120, -33, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 33*x^18 + 120*x^17 + 468*x^16 - 1392*x^15 - 3799*x^14 + 8012*x^13 + 19008*x^12 - 23720*x^11 - 57969*x^10 + 31228*x^9 + 101057*x^8 - 964*x^7 - 86922*x^6 - 33928*x^5 + 22258*x^4 + 19672*x^3 + 5396*x^2 + 600*x + 20)
 
gp: K = bnfinit(x^20 - 4*x^19 - 33*x^18 + 120*x^17 + 468*x^16 - 1392*x^15 - 3799*x^14 + 8012*x^13 + 19008*x^12 - 23720*x^11 - 57969*x^10 + 31228*x^9 + 101057*x^8 - 964*x^7 - 86922*x^6 - 33928*x^5 + 22258*x^4 + 19672*x^3 + 5396*x^2 + 600*x + 20, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 33 x^{18} + 120 x^{17} + 468 x^{16} - 1392 x^{15} - 3799 x^{14} + 8012 x^{13} + 19008 x^{12} - 23720 x^{11} - 57969 x^{10} + 31228 x^{9} + 101057 x^{8} - 964 x^{7} - 86922 x^{6} - 33928 x^{5} + 22258 x^{4} + 19672 x^{3} + 5396 x^{2} + 600 x + 20 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8501150111111046013911040000000000=2^{36}\cdot 5^{10}\cdot 103^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.71$
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:  $2, 5, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{82002810274962237907721334} a^{19} - \frac{2704405118015265707845201}{27334270091654079302573778} a^{18} + \frac{1364143770733060560334377}{9111423363884693100857926} a^{17} + \frac{1319449647594948000613637}{13667135045827039651286889} a^{16} - \frac{2922987935774800558340902}{13667135045827039651286889} a^{15} - \frac{7289816583162278664412249}{27334270091654079302573778} a^{14} - \frac{10232864096351809811257237}{82002810274962237907721334} a^{13} - \frac{7604729200655643438996869}{82002810274962237907721334} a^{12} - \frac{18185975503173042723510886}{41001405137481118953860667} a^{11} + \frac{2830885305059406383648489}{82002810274962237907721334} a^{10} - \frac{7217584424270156738228443}{82002810274962237907721334} a^{9} + \frac{2736115770984254691790803}{9111423363884693100857926} a^{8} - \frac{25731620155556499284397157}{82002810274962237907721334} a^{7} + \frac{9740710922358043928959391}{41001405137481118953860667} a^{6} - \frac{7940071345925794755435694}{41001405137481118953860667} a^{5} - \frac{8656629233345913912580459}{27334270091654079302573778} a^{4} - \frac{10612522750198688723149003}{41001405137481118953860667} a^{3} + \frac{2512423881699924966250516}{41001405137481118953860667} a^{2} - \frac{7510918346344883245665982}{41001405137481118953860667} a + \frac{19437422686195060676558690}{41001405137481118953860667}$

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:  $19$
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:  \( 70785617543.0 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), 10.10.3688067268608000.1

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

Sibling fields

Degree 5 sibling: 5.5.13579520.1
Degree 6 sibling: 6.6.1357952000.1
Degree 10 siblings: 10.10.3688067268608000.1, Deg 10
Degree 12 sibling: 12.12.1844033634304000000.1
Degree 15 sibling: Deg 15
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$2.8.16.8$x^{8} + 8 x^{5} + 12$$4$$2$$16$$S_4$$[8/3, 8/3]_{3}^{2}$
2.12.20.37$x^{12} - 6 x^{10} - x^{8} + 4 x^{6} + 3 x^{4} + 2 x^{2} - 7$$6$$2$$20$$S_4$$[8/3, 8/3]_{3}^{2}$
$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}$
$103$103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$