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Properties

Label 20.2.16156581965...0000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{20}\cdot 5^{10}\cdot 27517559^{5}$
Root discriminant $323.90$
Ramified primes $2, 5, 27517559$
Class number Not computed
Class group Not computed
Galois group 20T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-20836737422938446707879, 0, -1419239447710082479566, 0, -39929780990644264941, 0, -601902913742794431, 0, -5251386273851037, 0, -26248097666138, 0, -63291398882, 0, 17027900, 0, 511759, 0, 1215, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 1215*x^18 + 511759*x^16 + 17027900*x^14 - 63291398882*x^12 - 26248097666138*x^10 - 5251386273851037*x^8 - 601902913742794431*x^6 - 39929780990644264941*x^4 - 1419239447710082479566*x^2 - 20836737422938446707879)
 
gp: K = bnfinit(x^20 + 1215*x^18 + 511759*x^16 + 17027900*x^14 - 63291398882*x^12 - 26248097666138*x^10 - 5251386273851037*x^8 - 601902913742794431*x^6 - 39929780990644264941*x^4 - 1419239447710082479566*x^2 - 20836737422938446707879, 1)
 

Normalized defining polynomial

\( x^{20} + 1215 x^{18} + 511759 x^{16} + 17027900 x^{14} - 63291398882 x^{12} - 26248097666138 x^{10} - 5251386273851037 x^{8} - 601902913742794431 x^{6} - 39929780990644264941 x^{4} - 1419239447710082479566 x^{2} - 20836737422938446707879 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-161565819652412899832686999834204280821760000000000=-\,2^{20}\cdot 5^{10}\cdot 27517559^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $323.90$
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:  $2, 5, 27517559$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{27517559} a^{16} + \frac{1215}{27517559} a^{14} + \frac{511759}{27517559} a^{12} - \frac{10489659}{27517559} a^{10} - \frac{1013182}{27517559} a^{8} - \frac{6215485}{27517559} a^{6} - \frac{8072246}{27517559} a^{4} + \frac{1439300}{27517559} a^{2}$, $\frac{1}{27517559} a^{17} + \frac{1215}{27517559} a^{15} + \frac{511759}{27517559} a^{13} - \frac{10489659}{27517559} a^{11} - \frac{1013182}{27517559} a^{9} - \frac{6215485}{27517559} a^{7} - \frac{8072246}{27517559} a^{5} + \frac{1439300}{27517559} a^{3}$, $\frac{1}{1073446311430272990576859475028240101524751297451530564981437714850655571693063851419237} a^{18} - \frac{19350581563611523731252107655666006185122579647480911248296832686713556479142402}{1073446311430272990576859475028240101524751297451530564981437714850655571693063851419237} a^{16} - \frac{401271061441635127878487544985387737585450487048846003032063140029847216086027327483833}{1073446311430272990576859475028240101524751297451530564981437714850655571693063851419237} a^{14} + \frac{378135314252800535051625609843676281594186995529359501085982369198599360423433108492605}{1073446311430272990576859475028240101524751297451530564981437714850655571693063851419237} a^{12} - \frac{47200790408450861915490526237479746211732447783138579651447926709086563278439110559982}{1073446311430272990576859475028240101524751297451530564981437714850655571693063851419237} a^{10} + \frac{348178348746082101727162809102884525728528410436939858232871448189468959859114442521284}{1073446311430272990576859475028240101524751297451530564981437714850655571693063851419237} a^{8} + \frac{451358870916322194699744130386305877038340045141942666350930439651591263046894554305691}{1073446311430272990576859475028240101524751297451530564981437714850655571693063851419237} a^{6} + \frac{249615408863916472541302972050325706300493949028106162871742048787012899758830437102143}{1073446311430272990576859475028240101524751297451530564981437714850655571693063851419237} a^{4} - \frac{965680024531821710028513457279889900526403898551694929258753428743575653334277}{39009503402183056664904742278493528496650131556055919239836560897376673988163843} a^{2} - \frac{505316206377483710267712388633530413803734156701818229132296855961072454}{1417622231760566286599212607429806128394242074889561215798122242506200277}$, $\frac{1}{1073446311430272990576859475028240101524751297451530564981437714850655571693063851419237} a^{19} - \frac{19350581563611523731252107655666006185122579647480911248296832686713556479142402}{1073446311430272990576859475028240101524751297451530564981437714850655571693063851419237} a^{17} - \frac{401271061441635127878487544985387737585450487048846003032063140029847216086027327483833}{1073446311430272990576859475028240101524751297451530564981437714850655571693063851419237} a^{15} + \frac{378135314252800535051625609843676281594186995529359501085982369198599360423433108492605}{1073446311430272990576859475028240101524751297451530564981437714850655571693063851419237} a^{13} - \frac{47200790408450861915490526237479746211732447783138579651447926709086563278439110559982}{1073446311430272990576859475028240101524751297451530564981437714850655571693063851419237} a^{11} + \frac{348178348746082101727162809102884525728528410436939858232871448189468959859114442521284}{1073446311430272990576859475028240101524751297451530564981437714850655571693063851419237} a^{9} + \frac{451358870916322194699744130386305877038340045141942666350930439651591263046894554305691}{1073446311430272990576859475028240101524751297451530564981437714850655571693063851419237} a^{7} + \frac{249615408863916472541302972050325706300493949028106162871742048787012899758830437102143}{1073446311430272990576859475028240101524751297451530564981437714850655571693063851419237} a^{5} - \frac{965680024531821710028513457279889900526403898551694929258753428743575653334277}{39009503402183056664904742278493528496650131556055919239836560897376673988163843} a^{3} - \frac{505316206377483710267712388633530413803734156701818229132296855961072454}{1417622231760566286599212607429806128394242074889561215798122242506200277} a$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1022 are not computed
Character table for t20n1022 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.8.85992371875.1

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 32 sibling: 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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ $16{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
27517559Data not computed