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Properties

Label 20.8.94952935569...0625.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{14}\cdot 11^{5}\cdot 9931^{5}$
Root discriminant $56.09$
Ramified primes $5, 11, 9931$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1023

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![735025, -2493300, 3310850, -2524850, 1664625, -834850, -85475, 408860, -311905, 202230, -85235, 12810, 539, -3257, 2985, -932, 297, -112, 17, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 17*x^18 - 112*x^17 + 297*x^16 - 932*x^15 + 2985*x^14 - 3257*x^13 + 539*x^12 + 12810*x^11 - 85235*x^10 + 202230*x^9 - 311905*x^8 + 408860*x^7 - 85475*x^6 - 834850*x^5 + 1664625*x^4 - 2524850*x^3 + 3310850*x^2 - 2493300*x + 735025)
 
gp: K = bnfinit(x^20 - 4*x^19 + 17*x^18 - 112*x^17 + 297*x^16 - 932*x^15 + 2985*x^14 - 3257*x^13 + 539*x^12 + 12810*x^11 - 85235*x^10 + 202230*x^9 - 311905*x^8 + 408860*x^7 - 85475*x^6 - 834850*x^5 + 1664625*x^4 - 2524850*x^3 + 3310850*x^2 - 2493300*x + 735025, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 17 x^{18} - 112 x^{17} + 297 x^{16} - 932 x^{15} + 2985 x^{14} - 3257 x^{13} + 539 x^{12} + 12810 x^{11} - 85235 x^{10} + 202230 x^{9} - 311905 x^{8} + 408860 x^{7} - 85475 x^{6} - 834850 x^{5} + 1664625 x^{4} - 2524850 x^{3} + 3310850 x^{2} - 2493300 x + 735025 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(94952935569355941136243902587890625=5^{14}\cdot 11^{5}\cdot 9931^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.09$
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:  $5, 11, 9931$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{13} + \frac{2}{5} a^{12} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6}$, $\frac{1}{5} a^{17} + \frac{2}{5} a^{13} - \frac{2}{5} a^{12} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7}$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6}$, $\frac{1}{571927172574340459397642134722711808749139474824698405} a^{19} - \frac{31598015149999550923582307356152048481665806789426808}{571927172574340459397642134722711808749139474824698405} a^{18} - \frac{16521450882564353405172200974289482281405018538736132}{571927172574340459397642134722711808749139474824698405} a^{17} + \frac{19515113814013757050847932167334180930677558209919479}{571927172574340459397642134722711808749139474824698405} a^{16} + \frac{1700573175198924625540759331407275540274139659952089}{114385434514868091879528426944542361749827894964939681} a^{15} + \frac{7176253654803092881185062879239838578003203350728458}{571927172574340459397642134722711808749139474824698405} a^{14} - \frac{10190550866471628490934149381752347952797850406513909}{571927172574340459397642134722711808749139474824698405} a^{13} - \frac{112314357948038261990488909005220040640071801710096428}{571927172574340459397642134722711808749139474824698405} a^{12} + \frac{236936430756167570375112502323862773507247323553739298}{571927172574340459397642134722711808749139474824698405} a^{11} - \frac{25993492028922774121295562282292569443719676907246542}{114385434514868091879528426944542361749827894964939681} a^{10} + \frac{117020065496783629952166624718230833313158679946741704}{571927172574340459397642134722711808749139474824698405} a^{9} - \frac{99877385517562488107230519818767816092108872394621061}{571927172574340459397642134722711808749139474824698405} a^{8} - \frac{109342742308339979315751561444181634859853874689520649}{571927172574340459397642134722711808749139474824698405} a^{7} - \frac{129537763391978272382091136789731860203001159522572454}{571927172574340459397642134722711808749139474824698405} a^{6} + \frac{36931294788225722083252618845112439770969247628957491}{114385434514868091879528426944542361749827894964939681} a^{5} - \frac{35948096257871530195749380422109291038963044365267269}{114385434514868091879528426944542361749827894964939681} a^{4} + \frac{4002152507340366328280092599940735702403241773144893}{10398675864988008352684402449503851068166172269539971} a^{3} - \frac{3920201343558819936803767196156621334566829812179306}{114385434514868091879528426944542361749827894964939681} a^{2} + \frac{29914233681309578611542459077470462936577546746533914}{114385434514868091879528426944542361749827894964939681} a + \frac{57176546643437248432461151565268639959938410231454493}{114385434514868091879528426944542361749827894964939681}$

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:  $13$
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:  \( 13295880927.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.932312193828125.1

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ R $20$ R $20$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
9931Data not computed