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Properties

Label 20.20.3961078303...8125.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{15}\cdot 7^{10}\cdot 11^{16}$
Root discriminant $60.24$
Ramified primes $5, 7, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4621, -86946, 441152, -658628, -574064, 2013497, -355510, -2003579, 878982, 929822, -510719, -228288, 139797, 30774, -20399, -2224, 1609, 78, -64, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 64*x^18 + 78*x^17 + 1609*x^16 - 2224*x^15 - 20399*x^14 + 30774*x^13 + 139797*x^12 - 228288*x^11 - 510719*x^10 + 929822*x^9 + 878982*x^8 - 2003579*x^7 - 355510*x^6 + 2013497*x^5 - 574064*x^4 - 658628*x^3 + 441152*x^2 - 86946*x + 4621)
 
gp: K = bnfinit(x^20 - x^19 - 64*x^18 + 78*x^17 + 1609*x^16 - 2224*x^15 - 20399*x^14 + 30774*x^13 + 139797*x^12 - 228288*x^11 - 510719*x^10 + 929822*x^9 + 878982*x^8 - 2003579*x^7 - 355510*x^6 + 2013497*x^5 - 574064*x^4 - 658628*x^3 + 441152*x^2 - 86946*x + 4621, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 64 x^{18} + 78 x^{17} + 1609 x^{16} - 2224 x^{15} - 20399 x^{14} + 30774 x^{13} + 139797 x^{12} - 228288 x^{11} - 510719 x^{10} + 929822 x^{9} + 878982 x^{8} - 2003579 x^{7} - 355510 x^{6} + 2013497 x^{5} - 574064 x^{4} - 658628 x^{3} + 441152 x^{2} - 86946 x + 4621 \)

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:  \(396107830343483954099825714111328125=5^{15}\cdot 7^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.24$
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:  $5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(385=5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{385}(64,·)$, $\chi_{385}(1,·)$, $\chi_{385}(258,·)$, $\chi_{385}(71,·)$, $\chi_{385}(328,·)$, $\chi_{385}(202,·)$, $\chi_{385}(141,·)$, $\chi_{385}(342,·)$, $\chi_{385}(344,·)$, $\chi_{385}(27,·)$, $\chi_{385}(223,·)$, $\chi_{385}(97,·)$, $\chi_{385}(36,·)$, $\chi_{385}(169,·)$, $\chi_{385}(48,·)$, $\chi_{385}(309,·)$, $\chi_{385}(246,·)$, $\chi_{385}(377,·)$, $\chi_{385}(379,·)$, $\chi_{385}(188,·)$$\rbrace$
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}{53942677380093530752537159240814303273590603649} a^{19} - \frac{5259730474946546127379933827039225916865373465}{53942677380093530752537159240814303273590603649} a^{18} + \frac{11145132623772947388162363676301359717197132902}{53942677380093530752537159240814303273590603649} a^{17} - \frac{22922361421062346412037361246785262927217201364}{53942677380093530752537159240814303273590603649} a^{16} + \frac{4412005012727357638099139401150742484501525621}{53942677380093530752537159240814303273590603649} a^{15} + \frac{3121847717019942009849618280154332002565275550}{53942677380093530752537159240814303273590603649} a^{14} - \frac{14308995707477855056192999017815638251458388096}{53942677380093530752537159240814303273590603649} a^{13} - \frac{5749134105666280461343773570866095449985156842}{53942677380093530752537159240814303273590603649} a^{12} + \frac{9261191274149451525134482598642360994596644709}{53942677380093530752537159240814303273590603649} a^{11} + \frac{4118649099229782718608624118309689025923906558}{53942677380093530752537159240814303273590603649} a^{10} + \frac{11609116358463046297709315738526021579866191812}{53942677380093530752537159240814303273590603649} a^{9} - \frac{16353241027963414430247982026383079347032226834}{53942677380093530752537159240814303273590603649} a^{8} + \frac{5238707322474313765827705602474760148170857231}{53942677380093530752537159240814303273590603649} a^{7} - \frac{3380752620919236394310336076749411489629451390}{53942677380093530752537159240814303273590603649} a^{6} - \frac{9445436405578911077437725402698864591597546670}{53942677380093530752537159240814303273590603649} a^{5} + \frac{15261434775184239042780261082521572088030470041}{53942677380093530752537159240814303273590603649} a^{4} - \frac{19258921857588619237926646487010661761146724836}{53942677380093530752537159240814303273590603649} a^{3} + \frac{5861672644273236303042231410499627177602574973}{53942677380093530752537159240814303273590603649} a^{2} + \frac{9673013380028315421829892544818162991897430731}{53942677380093530752537159240814303273590603649} a - \frac{15035638178156743615732947294686135177313395664}{53942677380093530752537159240814303273590603649}$

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:  \( 249459949124 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.6125.1, \(\Q(\zeta_{11})^+\), 10.10.669871503125.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $20$ $20$ R R R $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
5Data not computed
7Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$