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Properties

Label 20.0.83086102921...3888.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 7^{8}\cdot 11^{8}\cdot 13^{9}$
Root discriminant $88.30$
Ramified primes $2, 3, 7, 11, 13$
Class number $1102$ (GRH)
Class group $[1102]$ (GRH)
Galois group $C_{10}^2:C_2^2$ (as 20T106)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2849617, 3955342, 2035604, 1360618, 8907493, 18362518, 18625392, 10877422, 4593593, 1381924, 399454, 63452, 35720, -8728, 1146, -1954, 45, -50, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 50*x^17 + 45*x^16 - 1954*x^15 + 1146*x^14 - 8728*x^13 + 35720*x^12 + 63452*x^11 + 399454*x^10 + 1381924*x^9 + 4593593*x^8 + 10877422*x^7 + 18625392*x^6 + 18362518*x^5 + 8907493*x^4 + 1360618*x^3 + 2035604*x^2 + 3955342*x + 2849617)
 
gp: K = bnfinit(x^20 - 50*x^17 + 45*x^16 - 1954*x^15 + 1146*x^14 - 8728*x^13 + 35720*x^12 + 63452*x^11 + 399454*x^10 + 1381924*x^9 + 4593593*x^8 + 10877422*x^7 + 18625392*x^6 + 18362518*x^5 + 8907493*x^4 + 1360618*x^3 + 2035604*x^2 + 3955342*x + 2849617, 1)
 

Normalized defining polynomial

\( x^{20} - 50 x^{17} + 45 x^{16} - 1954 x^{15} + 1146 x^{14} - 8728 x^{13} + 35720 x^{12} + 63452 x^{11} + 399454 x^{10} + 1381924 x^{9} + 4593593 x^{8} + 10877422 x^{7} + 18625392 x^{6} + 18362518 x^{5} + 8907493 x^{4} + 1360618 x^{3} + 2035604 x^{2} + 3955342 x + 2849617 \)

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:  \(830861029216363101082988400989215653888=2^{30}\cdot 3^{10}\cdot 7^{8}\cdot 11^{8}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.30$
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:  $2, 3, 7, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{719} a^{18} - \frac{90}{719} a^{17} + \frac{159}{719} a^{16} + \frac{24}{719} a^{15} - \frac{13}{719} a^{14} - \frac{114}{719} a^{13} + \frac{318}{719} a^{12} + \frac{93}{719} a^{11} - \frac{82}{719} a^{10} + \frac{270}{719} a^{9} + \frac{303}{719} a^{8} + \frac{46}{719} a^{7} - \frac{273}{719} a^{6} - \frac{239}{719} a^{5} - \frac{255}{719} a^{4} - \frac{342}{719} a^{3} - \frac{84}{719} a^{2} + \frac{80}{719} a - \frac{83}{719}$, $\frac{1}{33246407582992490958406266186377655651191930621787089551621142191174397071} a^{19} + \frac{13044653818394974594982444521160893763905583880796250367457011731211818}{33246407582992490958406266186377655651191930621787089551621142191174397071} a^{18} - \frac{14267572014665415498251257334602582729007395091420944110239303154750172847}{33246407582992490958406266186377655651191930621787089551621142191174397071} a^{17} + \frac{1189123826745729353685169994848110861391394577477236520023242520204757119}{33246407582992490958406266186377655651191930621787089551621142191174397071} a^{16} - \frac{2657462434805949403877947932632513702146733588829979344017267879370888625}{33246407582992490958406266186377655651191930621787089551621142191174397071} a^{15} - \frac{15125866478474141173977355103496332155080521044834230396548652333549464308}{33246407582992490958406266186377655651191930621787089551621142191174397071} a^{14} + \frac{15064191185642299832415918151626527494355691472457534476630724207845898451}{33246407582992490958406266186377655651191930621787089551621142191174397071} a^{13} + \frac{5074118807140869166082599513425362312977954099458762470726321100396186}{13389612397499996358600993228504895550218256392181671184704447116864437} a^{12} - \frac{12555255016038256798349789257356772198431006408068914015465833585451296306}{33246407582992490958406266186377655651191930621787089551621142191174397071} a^{11} + \frac{5136215539299109179573031864227564790522933052205516501924894117485023311}{33246407582992490958406266186377655651191930621787089551621142191174397071} a^{10} + \frac{1500626389453700046175524321229630076175939382577538475129587713488755580}{33246407582992490958406266186377655651191930621787089551621142191174397071} a^{9} - \frac{9371546578598218197008405053033537256906318570725304769597931289849812722}{33246407582992490958406266186377655651191930621787089551621142191174397071} a^{8} - \frac{15319834434621982814118567213007399982829825654216433716286423519489038494}{33246407582992490958406266186377655651191930621787089551621142191174397071} a^{7} + \frac{513397154398957218383591076371989189646621988007482181454602674832852470}{2557415967922499304492789706644435050091686970906699196278549399321107467} a^{6} - \frac{7526087123438979159842530389216767265907990383436824708909276080389664507}{33246407582992490958406266186377655651191930621787089551621142191174397071} a^{5} - \frac{3630736461616695906296605677106452270490705685321657949164480130477516848}{33246407582992490958406266186377655651191930621787089551621142191174397071} a^{4} + \frac{9056195529396246575904265605140236692781037946969488139424203382627161905}{33246407582992490958406266186377655651191930621787089551621142191174397071} a^{3} - \frac{875144043871515458935888075751871557216713873560413196177106069351274759}{2557415967922499304492789706644435050091686970906699196278549399321107467} a^{2} - \frac{15211752829419038781423022657707564825024825440104923330185062925719075774}{33246407582992490958406266186377655651191930621787089551621142191174397071} a + \frac{4599888301689809176604437949512558542327502008775278139981018828810286769}{33246407582992490958406266186377655651191930621787089551621142191174397071}$

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

Class group and class number

$C_{1102}$, which has order $1102$ (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:  \( 161877573.325 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{10}^2:C_2^2$ (as 20T106):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 46 conjugacy class representatives for $C_{10}^2:C_2^2$
Character table for $C_{10}^2:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.0.7488.1, 10.10.249828821987576832.1

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 R $20$ R R R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.10.8.2$x^{10} + 143 x^{5} + 5929$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$