↑

Properties

Label 20.0.10594318412...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{16}\cdot 5^{15}\cdot 73^{8}$
Root discriminant $44.80$
Ramified primes $3, 5, 73$
Class number $136$ (GRH)
Class group $[2, 68]$ (GRH)
Galois group $C_4\times A_5$ (as 20T63)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2025, -4050, 7560, -9315, 12456, -13500, 16869, -18390, 23095, -24435, 22236, -15010, 7959, -2865, 861, -180, 75, -20, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 9*x^18 - 20*x^17 + 75*x^16 - 180*x^15 + 861*x^14 - 2865*x^13 + 7959*x^12 - 15010*x^11 + 22236*x^10 - 24435*x^9 + 23095*x^8 - 18390*x^7 + 16869*x^6 - 13500*x^5 + 12456*x^4 - 9315*x^3 + 7560*x^2 - 4050*x + 2025)
 
gp: K = bnfinit(x^20 + 9*x^18 - 20*x^17 + 75*x^16 - 180*x^15 + 861*x^14 - 2865*x^13 + 7959*x^12 - 15010*x^11 + 22236*x^10 - 24435*x^9 + 23095*x^8 - 18390*x^7 + 16869*x^6 - 13500*x^5 + 12456*x^4 - 9315*x^3 + 7560*x^2 - 4050*x + 2025, 1)
 

Normalized defining polynomial

\( x^{20} + 9 x^{18} - 20 x^{17} + 75 x^{16} - 180 x^{15} + 861 x^{14} - 2865 x^{13} + 7959 x^{12} - 15010 x^{11} + 22236 x^{10} - 24435 x^{9} + 23095 x^{8} - 18390 x^{7} + 16869 x^{6} - 13500 x^{5} + 12456 x^{4} - 9315 x^{3} + 7560 x^{2} - 4050 x + 2025 \)

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:  \(1059431841229213450878936767578125=3^{16}\cdot 5^{15}\cdot 73^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.80$
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:  $3, 5, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{12} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{16} - \frac{2}{9} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{2}{9} a^{7} - \frac{1}{3} a^{6} + \frac{1}{9} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{2}{9} a^{8} - \frac{1}{3} a^{7} - \frac{2}{9} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{58290936474915} a^{18} + \frac{187727417788}{3886062431661} a^{17} - \frac{54141783484}{6476770719435} a^{16} - \frac{195102684121}{11658187294983} a^{15} - \frac{191151096872}{1295354143887} a^{14} + \frac{147905940569}{1295354143887} a^{13} - \frac{870831439663}{19430312158305} a^{12} - \frac{1769935774238}{3886062431661} a^{11} + \frac{1501097316643}{19430312158305} a^{10} + \frac{1026055060834}{11658187294983} a^{9} + \frac{833394487818}{2158923573145} a^{8} + \frac{52701014002}{431784714629} a^{7} + \frac{3965539948643}{11658187294983} a^{6} - \frac{1299400248271}{3886062431661} a^{5} + \frac{6430437489163}{19430312158305} a^{4} - \frac{355219420351}{1295354143887} a^{3} - \frac{702122031976}{6476770719435} a^{2} + \frac{67504460548}{431784714629} a - \frac{123477706781}{431784714629}$, $\frac{1}{74703790913683653310136085} a^{19} + \frac{28474390628}{24901263637894551103378695} a^{18} - \frac{454225020873980751915967}{24901263637894551103378695} a^{17} + \frac{32184049333819732663171}{74703790913683653310136085} a^{16} + \frac{76728164872530539855870}{553361414175434468963971} a^{15} - \frac{603872445655661675567393}{4980252727578910220675739} a^{14} - \frac{458593940725324647500303}{24901263637894551103378695} a^{13} + \frac{7925485049469919035666413}{24901263637894551103378695} a^{12} - \frac{10190041289229397549807922}{24901263637894551103378695} a^{11} + \frac{27050804198795340669159641}{74703790913683653310136085} a^{10} + \frac{603349885948185025236934}{8300421212631517034459565} a^{9} - \frac{9089953408180550993461037}{24901263637894551103378695} a^{8} - \frac{6752040681095424838908505}{14940758182736730662027217} a^{7} + \frac{380690284753612109602127}{4980252727578910220675739} a^{6} + \frac{461141644625984536426661}{8300421212631517034459565} a^{5} - \frac{8380193212394840695644583}{24901263637894551103378695} a^{4} - \frac{540961326086545366375031}{8300421212631517034459565} a^{3} + \frac{2644074980937218243113526}{8300421212631517034459565} a^{2} + \frac{127014160347978280585674}{553361414175434468963971} a - \frac{109994942765353322358522}{553361414175434468963971}$

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

Class group and class number

$C_{2}\times C_{68}$, which has order $136$ (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:  \( -\frac{3556016073550269494260}{4980252727578910220675739} a^{19} + \frac{6680883908153776147006}{74703790913683653310136085} a^{18} - \frac{28843958497103237858908}{4980252727578910220675739} a^{17} + \frac{128411824032816501447181}{8300421212631517034459565} a^{16} - \frac{741220817104442143114900}{14940758182736730662027217} a^{15} + \frac{626272302684405051788158}{4980252727578910220675739} a^{14} - \frac{2945030537318227771759202}{4980252727578910220675739} a^{13} + \frac{50657238374946524890926662}{24901263637894551103378695} a^{12} - \frac{27191470947287135392428533}{4980252727578910220675739} a^{11} + \frac{82415707707200303191041706}{8300421212631517034459565} a^{10} - \frac{197778563483926553549384654}{14940758182736730662027217} a^{9} + \frac{318076426567038580328941222}{24901263637894551103378695} a^{8} - \frac{49000518039101067388324382}{4980252727578910220675739} a^{7} + \frac{106517731686079640086724750}{14940758182736730662027217} a^{6} - \frac{11886068175155009121469892}{1660084242526303406891913} a^{5} + \frac{19288116027245964094935792}{2766807070877172344819855} a^{4} - \frac{2963346511990376214878906}{553361414175434468963971} a^{3} + \frac{12921178226406221510798568}{2766807070877172344819855} a^{2} - \frac{1525998162249812542461747}{553361414175434468963971} a + \frac{933816872071413534966552}{553361414175434468963971} \) (order $10$)
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:  \( 6959926.34763 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times A_5$ (as 20T63):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 240
The 20 conjugacy class representatives for $C_4\times A_5$
Character table for $C_4\times A_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.10791225.1, 10.10.582252685003125.1

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

Sibling fields

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 $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ $20$ $20$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.12.12.23$x^{12} + 21 x^{11} + 21 x^{10} + 63 x^{9} + 36 x^{8} + 54 x^{7} + 90 x^{6} + 81 x^{3} - 81$$3$$4$$12$$S_3 \times C_4$$[3/2]_{2}^{4}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$73$73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
73.12.8.2$x^{12} - 389017 x^{3} + 369177133$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$