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Properties

Label 20.20.1935602494...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 5^{10}\cdot 17^{6}\cdot 23^{8}$
Root discriminant $36.67$
Ramified primes $2, 5, 17, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times S_5$ (as 20T62)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -140, -634, 832, 5588, 300, -16182, -5240, 23081, 8580, -18596, -6324, 8869, 2434, -2496, -494, 396, 50, -32, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 32*x^18 + 50*x^17 + 396*x^16 - 494*x^15 - 2496*x^14 + 2434*x^13 + 8869*x^12 - 6324*x^11 - 18596*x^10 + 8580*x^9 + 23081*x^8 - 5240*x^7 - 16182*x^6 + 300*x^5 + 5588*x^4 + 832*x^3 - 634*x^2 - 140*x + 5)
 
gp: K = bnfinit(x^20 - 2*x^19 - 32*x^18 + 50*x^17 + 396*x^16 - 494*x^15 - 2496*x^14 + 2434*x^13 + 8869*x^12 - 6324*x^11 - 18596*x^10 + 8580*x^9 + 23081*x^8 - 5240*x^7 - 16182*x^6 + 300*x^5 + 5588*x^4 + 832*x^3 - 634*x^2 - 140*x + 5, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 32 x^{18} + 50 x^{17} + 396 x^{16} - 494 x^{15} - 2496 x^{14} + 2434 x^{13} + 8869 x^{12} - 6324 x^{11} - 18596 x^{10} + 8580 x^{9} + 23081 x^{8} - 5240 x^{7} - 16182 x^{6} + 300 x^{5} + 5588 x^{4} + 832 x^{3} - 634 x^{2} - 140 x + 5 \)

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:  \(19356024941343968143360000000000=2^{20}\cdot 5^{10}\cdot 17^{6}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.67$
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:  $2, 5, 17, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{46} a^{18} - \frac{3}{46} a^{17} + \frac{11}{46} a^{16} + \frac{11}{46} a^{15} - \frac{3}{46} a^{14} + \frac{9}{23} a^{13} + \frac{11}{46} a^{12} - \frac{4}{23} a^{11} + \frac{1}{23} a^{10} - \frac{11}{23} a^{9} - \frac{1}{23} a^{8} + \frac{10}{23} a^{7} - \frac{19}{46} a^{6} - \frac{5}{46} a^{5} - \frac{9}{46} a^{4} + \frac{17}{46} a^{3} + \frac{13}{46} a^{2} + \frac{2}{23} a - \frac{3}{46}$, $\frac{1}{77196243384278668} a^{19} - \frac{366547822062271}{77196243384278668} a^{18} + \frac{18012395880774031}{77196243384278668} a^{17} - \frac{18012626021641605}{77196243384278668} a^{16} + \frac{554451304071647}{3356358408012116} a^{15} + \frac{13035986789841633}{77196243384278668} a^{14} - \frac{18605515922173903}{77196243384278668} a^{13} + \frac{25649190657785241}{77196243384278668} a^{12} + \frac{1040609842058865}{2969086284010718} a^{11} + \frac{10802577886577045}{38598121692139334} a^{10} - \frac{7697650617516477}{38598121692139334} a^{9} + \frac{8088579450410857}{38598121692139334} a^{8} - \frac{37277382296355581}{77196243384278668} a^{7} - \frac{19448372408114655}{77196243384278668} a^{6} - \frac{32208619484795203}{77196243384278668} a^{5} + \frac{3577849719466359}{77196243384278668} a^{4} + \frac{19236836017380333}{77196243384278668} a^{3} - \frac{11490852585378221}{77196243384278668} a^{2} + \frac{5176852206811869}{77196243384278668} a - \frac{12575104714347689}{77196243384278668}$

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:  \( 2648617715.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_5$ (as 20T62):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 240
The 14 conjugacy class representatives for $C_2\times S_5$
Character table for $C_2\times S_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.175981930624000.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 12 siblings: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 30 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$2.8.8.11$x^{8} + 20 x^{2} + 4$$4$$2$$8$$S_4$$[4/3, 4/3]_{3}^{2}$
2.12.12.28$x^{12} - x^{10} + 2 x^{8} - x^{6} - 2 x^{4} + 3 x^{2} + 1$$6$$2$$12$$S_4$$[4/3, 4/3]_{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}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.12.6.1$x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$