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Properties

Label 20.20.6197891644...3125.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{10}\cdot 5^{15}\cdot 61^{4}\cdot 397^{4}$
Root discriminant $43.61$
Ramified primes $3, 5, 61, 397$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\times S_5$ (as 20T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![61, 732, -1317, -7316, 9352, 27081, -29261, -47573, 44365, 44358, -35526, -23105, 15654, 6814, -3841, -1110, 519, 92, -36, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 36*x^18 + 92*x^17 + 519*x^16 - 1110*x^15 - 3841*x^14 + 6814*x^13 + 15654*x^12 - 23105*x^11 - 35526*x^10 + 44358*x^9 + 44365*x^8 - 47573*x^7 - 29261*x^6 + 27081*x^5 + 9352*x^4 - 7316*x^3 - 1317*x^2 + 732*x + 61)
 
gp: K = bnfinit(x^20 - 3*x^19 - 36*x^18 + 92*x^17 + 519*x^16 - 1110*x^15 - 3841*x^14 + 6814*x^13 + 15654*x^12 - 23105*x^11 - 35526*x^10 + 44358*x^9 + 44365*x^8 - 47573*x^7 - 29261*x^6 + 27081*x^5 + 9352*x^4 - 7316*x^3 - 1317*x^2 + 732*x + 61, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 36 x^{18} + 92 x^{17} + 519 x^{16} - 1110 x^{15} - 3841 x^{14} + 6814 x^{13} + 15654 x^{12} - 23105 x^{11} - 35526 x^{10} + 44358 x^{9} + 44365 x^{8} - 47573 x^{7} - 29261 x^{6} + 27081 x^{5} + 9352 x^{4} - 7316 x^{3} - 1317 x^{2} + 732 x + 61 \)

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:  \(619789164416171417636993408203125=3^{10}\cdot 5^{15}\cdot 61^{4}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.61$
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:  $3, 5, 61, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{11417739369188919722266553891} a^{19} + \frac{1743453126750418665417641261}{11417739369188919722266553891} a^{18} + \frac{4776769134428740342729053266}{11417739369188919722266553891} a^{17} - \frac{3062857175436958593391994315}{11417739369188919722266553891} a^{16} + \frac{2138954085520003569682484943}{11417739369188919722266553891} a^{15} + \frac{163514043770587488262657798}{1037976306289901792933323081} a^{14} - \frac{3260228159236732459163575200}{11417739369188919722266553891} a^{13} + \frac{377521942431606263111043767}{1037976306289901792933323081} a^{12} + \frac{1060251243817792739237104922}{11417739369188919722266553891} a^{11} + \frac{1794401185381946842963677396}{11417739369188919722266553891} a^{10} + \frac{1106348229554943323390384674}{11417739369188919722266553891} a^{9} - \frac{1228664095594777897448080315}{11417739369188919722266553891} a^{8} + \frac{3123566701393104216586353847}{11417739369188919722266553891} a^{7} - \frac{5459588157603633979055967168}{11417739369188919722266553891} a^{6} + \frac{778780532641947159092516115}{11417739369188919722266553891} a^{5} - \frac{1337572349662648328984948451}{11417739369188919722266553891} a^{4} - \frac{5072176436035603207019417573}{11417739369188919722266553891} a^{3} - \frac{4390741157363284426486354656}{11417739369188919722266553891} a^{2} + \frac{23171708859501155617150673}{187176055232605241348632031} a - \frac{7923507944854541278231890}{187176055232605241348632031}$

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:  \( 10094804779.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times S_5$ (as 20T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 480
The 28 conjugacy class representatives for $C_4\times S_5$
Character table for $C_4\times S_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 5.5.24217.1, 10.10.1832697153125.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 24 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 $20$ R R $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
3Data not computed
5Data not computed
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
397Data not computed