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Properties

Label 20.0.30210858588...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 31^{2}\cdot 409^{5}\cdot 167711^{2}$
Root discriminant $47.21$
Ramified primes $5, 31, 409, 167711$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group 20T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![35645021, 12803557, 27779804, 14203125, 16216658, 4248851, 8534495, -451586, 3345087, -628003, 880244, -187172, 153043, -29493, 17378, -2692, 1247, -135, 52, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 52*x^18 - 135*x^17 + 1247*x^16 - 2692*x^15 + 17378*x^14 - 29493*x^13 + 153043*x^12 - 187172*x^11 + 880244*x^10 - 628003*x^9 + 3345087*x^8 - 451586*x^7 + 8534495*x^6 + 4248851*x^5 + 16216658*x^4 + 14203125*x^3 + 27779804*x^2 + 12803557*x + 35645021)
 
gp: K = bnfinit(x^20 - 3*x^19 + 52*x^18 - 135*x^17 + 1247*x^16 - 2692*x^15 + 17378*x^14 - 29493*x^13 + 153043*x^12 - 187172*x^11 + 880244*x^10 - 628003*x^9 + 3345087*x^8 - 451586*x^7 + 8534495*x^6 + 4248851*x^5 + 16216658*x^4 + 14203125*x^3 + 27779804*x^2 + 12803557*x + 35645021, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 52 x^{18} - 135 x^{17} + 1247 x^{16} - 2692 x^{15} + 17378 x^{14} - 29493 x^{13} + 153043 x^{12} - 187172 x^{11} + 880244 x^{10} - 628003 x^{9} + 3345087 x^{8} - 451586 x^{7} + 8534495 x^{6} + 4248851 x^{5} + 16216658 x^{4} + 14203125 x^{3} + 27779804 x^{2} + 12803557 x + 35645021 \)

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:  \(3021085858887094365354739931640625=5^{10}\cdot 31^{2}\cdot 409^{5}\cdot 167711^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.21$
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:  $5, 31, 409, 167711$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{101} a^{18} - \frac{11}{101} a^{17} - \frac{24}{101} a^{16} + \frac{43}{101} a^{15} - \frac{9}{101} a^{14} + \frac{24}{101} a^{13} - \frac{23}{101} a^{12} - \frac{16}{101} a^{11} - \frac{11}{101} a^{10} - \frac{34}{101} a^{9} - \frac{16}{101} a^{8} - \frac{38}{101} a^{7} - \frac{34}{101} a^{6} + \frac{25}{101} a^{5} + \frac{18}{101} a^{4} - \frac{19}{101} a^{3} + \frac{25}{101} a^{2} - \frac{13}{101} a$, $\frac{1}{1941492526683626081553446914784196303239988715356740958446263} a^{19} + \frac{7169209258693361607308165688491846210535505673007733510062}{1941492526683626081553446914784196303239988715356740958446263} a^{18} + \frac{47975227037674439938132120447114702695721764202669126761351}{1941492526683626081553446914784196303239988715356740958446263} a^{17} + \frac{192836680106650961976512427016291062631629998415994758861206}{1941492526683626081553446914784196303239988715356740958446263} a^{16} - \frac{47766652664361157180290410310111192778797742270106394917710}{1941492526683626081553446914784196303239988715356740958446263} a^{15} - \frac{494304398607470552259671580643627747443925589807560701623549}{1941492526683626081553446914784196303239988715356740958446263} a^{14} - \frac{915650752552079554215545361869121874833621586896153082633448}{1941492526683626081553446914784196303239988715356740958446263} a^{13} + \frac{104531974507532141822957712306856753085502189777379553594048}{1941492526683626081553446914784196303239988715356740958446263} a^{12} - \frac{513152681927779630019814978741578845123256184565912780025195}{1941492526683626081553446914784196303239988715356740958446263} a^{11} + \frac{939265718249229747794512376232299110541601248760281521140972}{1941492526683626081553446914784196303239988715356740958446263} a^{10} - \frac{65777281233688202698994360812105413181260583029983017608024}{1941492526683626081553446914784196303239988715356740958446263} a^{9} + \frac{838812345522104728206135320751182259998362832889284214012880}{1941492526683626081553446914784196303239988715356740958446263} a^{8} + \frac{797868199784417431797196033776772414544450533252095624923886}{1941492526683626081553446914784196303239988715356740958446263} a^{7} + \frac{43035052514911764065644361765926163107814317898423125642735}{1941492526683626081553446914784196303239988715356740958446263} a^{6} - \frac{654228172027629724426167247022013803448428156055662158308343}{1941492526683626081553446914784196303239988715356740958446263} a^{5} - \frac{470558245327570073681622853684350433448425414791447341206234}{1941492526683626081553446914784196303239988715356740958446263} a^{4} - \frac{771427275282982274333431924816400165802615328544990187654385}{1941492526683626081553446914784196303239988715356740958446263} a^{3} + \frac{945094328533609482237994431674124052654580184388170598789745}{1941492526683626081553446914784196303239988715356740958446263} a^{2} + \frac{576734089531161807471299356154301103295928150535637923268704}{1941492526683626081553446914784196303239988715356740958446263} a - \frac{1773610397514317069794957262598008009362802577843126623467}{19222698283996297837162840740437587160791967478779613449963}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( -1 \) (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:  \( 31988778.9557 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 115200
The 119 conjugacy class representatives for t20n781 are not computed
Character table for t20n781 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.10225.1, 10.6.16247003125.1

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ $20$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
409Data not computed
167711Data not computed