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Properties

Label 20.0.14154027166...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{15}\cdot 89^{7}$
Root discriminant $32.18$
Ramified primes $2, 5, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T658

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2501, 1565, 8131, 175, 6223, -728, 11130, -13611, 13021, -10821, 9694, -6743, 3509, -1298, 690, -441, 157, -39, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 21*x^18 - 39*x^17 + 157*x^16 - 441*x^15 + 690*x^14 - 1298*x^13 + 3509*x^12 - 6743*x^11 + 9694*x^10 - 10821*x^9 + 13021*x^8 - 13611*x^7 + 11130*x^6 - 728*x^5 + 6223*x^4 + 175*x^3 + 8131*x^2 + 1565*x + 2501)
 
gp: K = bnfinit(x^20 - 6*x^19 + 21*x^18 - 39*x^17 + 157*x^16 - 441*x^15 + 690*x^14 - 1298*x^13 + 3509*x^12 - 6743*x^11 + 9694*x^10 - 10821*x^9 + 13021*x^8 - 13611*x^7 + 11130*x^6 - 728*x^5 + 6223*x^4 + 175*x^3 + 8131*x^2 + 1565*x + 2501, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 21 x^{18} - 39 x^{17} + 157 x^{16} - 441 x^{15} + 690 x^{14} - 1298 x^{13} + 3509 x^{12} - 6743 x^{11} + 9694 x^{10} - 10821 x^{9} + 13021 x^{8} - 13611 x^{7} + 11130 x^{6} - 728 x^{5} + 6223 x^{4} + 175 x^{3} + 8131 x^{2} + 1565 x + 2501 \)

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:  \(1415402716656928000000000000000=2^{20}\cdot 5^{15}\cdot 89^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.18$
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:  $2, 5, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{10} a^{16} - \frac{1}{2} a^{15} - \frac{3}{10} a^{13} + \frac{3}{10} a^{12} - \frac{1}{2} a^{11} - \frac{1}{10} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{10} a^{6} + \frac{2}{5} a^{5} + \frac{3}{10} a^{4} - \frac{2}{5} a^{3} + \frac{1}{10} a^{2} + \frac{1}{10} a + \frac{1}{10}$, $\frac{1}{10} a^{17} - \frac{1}{2} a^{15} - \frac{3}{10} a^{14} - \frac{1}{5} a^{13} + \frac{2}{5} a^{11} - \frac{3}{10} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{10} a^{7} - \frac{1}{10} a^{6} + \frac{3}{10} a^{5} + \frac{1}{10} a^{4} + \frac{1}{10} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{10} a^{18} + \frac{1}{5} a^{15} - \frac{1}{5} a^{14} - \frac{1}{2} a^{13} - \frac{1}{10} a^{12} + \frac{1}{5} a^{11} - \frac{1}{10} a^{10} - \frac{1}{5} a^{9} + \frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{1}{5} a^{6} + \frac{1}{10} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2}$, $\frac{1}{2470652153372640619059664933729533553765887590} a^{19} - \frac{13310567519661447121677852686076765756572235}{494130430674528123811932986745906710753177518} a^{18} + \frac{53109691076162283241370720686799907949558563}{1235326076686320309529832466864766776882943795} a^{17} + \frac{22685107385393069157680180755854237518711953}{2470652153372640619059664933729533553765887590} a^{16} + \frac{787295109076457586669800630630001457815547823}{2470652153372640619059664933729533553765887590} a^{15} + \frac{370230359369650742112641730464477353442206377}{2470652153372640619059664933729533553765887590} a^{14} - \frac{50690612670511811421708985938249845867820091}{2470652153372640619059664933729533553765887590} a^{13} + \frac{95377026467234185101090579025542519400026005}{247065215337264061905966493372953355376588759} a^{12} + \frac{520617525473524935980923023690284601337915029}{1235326076686320309529832466864766776882943795} a^{11} - \frac{146574561413176009071134553251720356439962918}{1235326076686320309529832466864766776882943795} a^{10} - \frac{573274761125923422107726483891132372919705993}{2470652153372640619059664933729533553765887590} a^{9} - \frac{433234000911126238994492932359363693319694517}{1235326076686320309529832466864766776882943795} a^{8} + \frac{142391518178758604757246845960119093347345037}{2470652153372640619059664933729533553765887590} a^{7} - \frac{383002589535225380347582301224717654404946122}{1235326076686320309529832466864766776882943795} a^{6} - \frac{285306334372611222065369914862822479419671407}{2470652153372640619059664933729533553765887590} a^{5} + \frac{8961488036367506289619335950966583870050437}{494130430674528123811932986745906710753177518} a^{4} + \frac{1103512640818550141173069170749334908799770933}{2470652153372640619059664933729533553765887590} a^{3} - \frac{103803495895662621843687253364997089532752024}{1235326076686320309529832466864766776882943795} a^{2} + \frac{16328566099259808908843202267213079859902796}{1235326076686320309529832466864766776882943795} a + \frac{294292292450660323905674800267781436635185983}{1235326076686320309529832466864766776882943795}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 2452904.06156 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T658:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 76 conjugacy class representatives for t20n658 are not computed
Character table for t20n658 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.11125.1, 10.2.25347200000.1

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

Sibling fields

Degree 20 siblings: 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
89Data not computed