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Properties

Label 20.4.65846307453...0000.2
Degree $20$
Signature $[4, 8]$
Discriminant $2^{28}\cdot 5^{11}\cdot 3469^{5}$
Root discriminant $49.08$
Ramified primes $2, 5, 3469$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![154804, -516048, 716304, -761240, 821652, -630084, 363896, -186004, 48952, 25800, -41810, 36028, -23702, 12292, -5296, 1834, -468, 66, 12, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 12*x^18 + 66*x^17 - 468*x^16 + 1834*x^15 - 5296*x^14 + 12292*x^13 - 23702*x^12 + 36028*x^11 - 41810*x^10 + 25800*x^9 + 48952*x^8 - 186004*x^7 + 363896*x^6 - 630084*x^5 + 821652*x^4 - 761240*x^3 + 716304*x^2 - 516048*x + 154804)
 
gp: K = bnfinit(x^20 - 8*x^19 + 12*x^18 + 66*x^17 - 468*x^16 + 1834*x^15 - 5296*x^14 + 12292*x^13 - 23702*x^12 + 36028*x^11 - 41810*x^10 + 25800*x^9 + 48952*x^8 - 186004*x^7 + 363896*x^6 - 630084*x^5 + 821652*x^4 - 761240*x^3 + 716304*x^2 - 516048*x + 154804, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 12 x^{18} + 66 x^{17} - 468 x^{16} + 1834 x^{15} - 5296 x^{14} + 12292 x^{13} - 23702 x^{12} + 36028 x^{11} - 41810 x^{10} + 25800 x^{9} + 48952 x^{8} - 186004 x^{7} + 363896 x^{6} - 630084 x^{5} + 821652 x^{4} - 761240 x^{3} + 716304 x^{2} - 516048 x + 154804 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6584630745307392888012800000000000=2^{28}\cdot 5^{11}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.08$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{2} a^{16}$, $\frac{1}{2} a^{17}$, $\frac{1}{2} a^{18}$, $\frac{1}{3238886886858208744987830969405613824893211292692562} a^{19} - \frac{557286470091871730956555673124035013651130657117811}{3238886886858208744987830969405613824893211292692562} a^{18} - \frac{493458103510286689250668985306902750588752718631583}{3238886886858208744987830969405613824893211292692562} a^{17} + \frac{785050944663620311217087245455429989522015315804793}{3238886886858208744987830969405613824893211292692562} a^{16} + \frac{20761545185391269786559425263252717933327397072199}{249145145142939134229833151492739524991785484053274} a^{15} + \frac{151894935549410724153116331121798799414018255367564}{1619443443429104372493915484702806912446605646346281} a^{14} + \frac{123653566199951823929935644797665286569181164059093}{1619443443429104372493915484702806912446605646346281} a^{13} + \frac{218848643396898022707315935806503132019010155922709}{1619443443429104372493915484702806912446605646346281} a^{12} + \frac{351371956859079170892736660277602937318233994866673}{3238886886858208744987830969405613824893211292692562} a^{11} - \frac{386652667739754049193898080283107136340176306171175}{3238886886858208744987830969405613824893211292692562} a^{10} - \frac{688757503464697056764071622214827273470359579044365}{1619443443429104372493915484702806912446605646346281} a^{9} + \frac{313474901634960444453806514441385406410360874046118}{1619443443429104372493915484702806912446605646346281} a^{8} + \frac{179850695963890383820759205248353550095581443700389}{1619443443429104372493915484702806912446605646346281} a^{7} + \frac{30155192511360781733101899422138693908730880658303}{124572572571469567114916575746369762495892742026637} a^{6} - \frac{30852944100454438462259726782790626724868699566542}{124572572571469567114916575746369762495892742026637} a^{5} - \frac{1067430160326290425085943694714700273949952760140}{124572572571469567114916575746369762495892742026637} a^{4} + \frac{19180358631040809404619163560392444921551779904553}{124572572571469567114916575746369762495892742026637} a^{3} - \frac{114718841300167736241069612771716504637203259928349}{1619443443429104372493915484702806912446605646346281} a^{2} - \frac{368178130158022092445400529913864976545714398007291}{1619443443429104372493915484702806912446605646346281} a + \frac{13527614127817982538115296795665647283234643100696}{124572572571469567114916575746369762495892742026637}$

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:  $11$
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:  \( 646212999.983 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 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 $20$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3469Data not computed