↑

Properties

Label 20.20.1192035046...9392.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{16}\cdot 11^{12}\cdot 157^{9}$
Root discriminant $71.42$
Ramified primes $2, 11, 157$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_5$ (as 20T35)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-18511, 9590, 181022, -109710, -525839, 412832, 620232, -598132, -344106, 427696, 84220, -166956, -1906, 36624, -3160, -4412, 601, 266, -42, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 42*x^18 + 266*x^17 + 601*x^16 - 4412*x^15 - 3160*x^14 + 36624*x^13 - 1906*x^12 - 166956*x^11 + 84220*x^10 + 427696*x^9 - 344106*x^8 - 598132*x^7 + 620232*x^6 + 412832*x^5 - 525839*x^4 - 109710*x^3 + 181022*x^2 + 9590*x - 18511)
 
gp: K = bnfinit(x^20 - 6*x^19 - 42*x^18 + 266*x^17 + 601*x^16 - 4412*x^15 - 3160*x^14 + 36624*x^13 - 1906*x^12 - 166956*x^11 + 84220*x^10 + 427696*x^9 - 344106*x^8 - 598132*x^7 + 620232*x^6 + 412832*x^5 - 525839*x^4 - 109710*x^3 + 181022*x^2 + 9590*x - 18511, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 42 x^{18} + 266 x^{17} + 601 x^{16} - 4412 x^{15} - 3160 x^{14} + 36624 x^{13} - 1906 x^{12} - 166956 x^{11} + 84220 x^{10} + 427696 x^{9} - 344106 x^{8} - 598132 x^{7} + 620232 x^{6} + 412832 x^{5} - 525839 x^{4} - 109710 x^{3} + 181022 x^{2} + 9590 x - 18511 \)

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:  \(11920350468069903076814830978072379392=2^{16}\cdot 11^{12}\cdot 157^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.42$
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:  $2, 11, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{3}{8} a^{4} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{8} a^{16} - \frac{1}{4} a^{8} + \frac{1}{8}$, $\frac{1}{1144} a^{17} + \frac{3}{286} a^{16} + \frac{3}{52} a^{15} - \frac{5}{104} a^{14} + \frac{10}{143} a^{13} - \frac{79}{1144} a^{12} + \frac{3}{52} a^{11} - \frac{1}{104} a^{10} + \frac{2}{143} a^{9} - \frac{73}{1144} a^{8} - \frac{67}{572} a^{7} - \frac{241}{1144} a^{6} - \frac{31}{143} a^{5} - \frac{193}{1144} a^{4} - \frac{267}{572} a^{3} + \frac{555}{1144} a^{2} + \frac{63}{1144} a - \frac{291}{1144}$, $\frac{1}{14872} a^{18} + \frac{3}{7436} a^{17} - \frac{3}{7436} a^{16} - \frac{1}{676} a^{15} + \frac{553}{14872} a^{14} - \frac{27}{572} a^{13} + \frac{111}{14872} a^{12} + \frac{1}{676} a^{11} - \frac{61}{14872} a^{10} + \frac{49}{286} a^{9} + \frac{3593}{14872} a^{8} + \frac{1783}{7436} a^{7} - \frac{1233}{14872} a^{6} - \frac{711}{7436} a^{5} + \frac{477}{1144} a^{4} + \frac{3381}{7436} a^{3} - \frac{97}{338} a^{2} - \frac{1407}{7436} a - \frac{2115}{14872}$, $\frac{1}{173125630312485136} a^{19} - \frac{206908636471}{13317356177883472} a^{18} - \frac{5729678900897}{173125630312485136} a^{17} + \frac{10409134489614511}{173125630312485136} a^{16} + \frac{1347765545918395}{21640703789060642} a^{15} + \frac{4213452503261175}{86562815156242568} a^{14} - \frac{363752096929243}{3934673416192844} a^{13} - \frac{7895768927205117}{86562815156242568} a^{12} - \frac{8489489537352563}{86562815156242568} a^{11} + \frac{849429620379423}{10820351894530321} a^{10} + \frac{20741747258989987}{86562815156242568} a^{9} + \frac{15837967726985}{128051501710418} a^{8} - \frac{182285180770514}{10820351894530321} a^{7} - \frac{116092010950109}{7869346832385688} a^{6} + \frac{6311411428331517}{43281407578121284} a^{5} + \frac{40979502163402609}{86562815156242568} a^{4} + \frac{4084300739530903}{15738693664771376} a^{3} - \frac{74273832091330997}{173125630312485136} a^{2} + \frac{63295197322075291}{173125630312485136} a + \frac{73514774312838497}{173125630312485136}$

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:  \( 2864940973830 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_5$ (as 20T35):

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

10.10.1755072387321088.1

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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$157$$\Q_{157}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{157}$$x + 5$$1$$1$$0$Trivial$[\ ]$
157.6.3.1$x^{6} - 314 x^{4} + 24649 x^{2} - 870725925$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
157.6.3.1$x^{6} - 314 x^{4} + 24649 x^{2} - 870725925$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
157.6.3.1$x^{6} - 314 x^{4} + 24649 x^{2} - 870725925$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$