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Properties

Label 20.0.21125000000...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{22}\cdot 5^{25}\cdot 13^{2}$
Root discriminant $20.71$
Ramified primes $2, 5, 13$
Class number $1$
Class group Trivial
Galois group 20T138

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10, 0, 30, -30, 35, -82, -15, 40, 5, 160, -270, -40, 605, -960, 930, -664, 370, -160, 50, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 50*x^18 - 160*x^17 + 370*x^16 - 664*x^15 + 930*x^14 - 960*x^13 + 605*x^12 - 40*x^11 - 270*x^10 + 160*x^9 + 5*x^8 + 40*x^7 - 15*x^6 - 82*x^5 + 35*x^4 - 30*x^3 + 30*x^2 + 10)
 
gp: K = bnfinit(x^20 - 10*x^19 + 50*x^18 - 160*x^17 + 370*x^16 - 664*x^15 + 930*x^14 - 960*x^13 + 605*x^12 - 40*x^11 - 270*x^10 + 160*x^9 + 5*x^8 + 40*x^7 - 15*x^6 - 82*x^5 + 35*x^4 - 30*x^3 + 30*x^2 + 10, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 50 x^{18} - 160 x^{17} + 370 x^{16} - 664 x^{15} + 930 x^{14} - 960 x^{13} + 605 x^{12} - 40 x^{11} - 270 x^{10} + 160 x^{9} + 5 x^{8} + 40 x^{7} - 15 x^{6} - 82 x^{5} + 35 x^{4} - 30 x^{3} + 30 x^{2} + 10 \)

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:  \(211250000000000000000000000=2^{22}\cdot 5^{25}\cdot 13^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.71$
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:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{6198313388500875118451} a^{19} - \frac{2369994515387737805241}{6198313388500875118451} a^{18} - \frac{24468708840871700253}{6198313388500875118451} a^{17} + \frac{1090123887545899500792}{6198313388500875118451} a^{16} - \frac{2476111539336502795899}{6198313388500875118451} a^{15} + \frac{2077880641451604188579}{6198313388500875118451} a^{14} + \frac{470725780068541637654}{6198313388500875118451} a^{13} + \frac{73331851490335435516}{476793337576990393727} a^{12} + \frac{325239367348328829884}{6198313388500875118451} a^{11} + \frac{1102022362807346886020}{6198313388500875118451} a^{10} + \frac{2630953342961734382326}{6198313388500875118451} a^{9} + \frac{1247107437589018552940}{6198313388500875118451} a^{8} - \frac{216648371081684762117}{6198313388500875118451} a^{7} - \frac{2457918399144832720496}{6198313388500875118451} a^{6} - \frac{2478476713229349330298}{6198313388500875118451} a^{5} - \frac{2224184105303443477234}{6198313388500875118451} a^{4} - \frac{1232167117523659334149}{6198313388500875118451} a^{3} - \frac{5062023168215311039}{12886306420999740371} a^{2} - \frac{2832692395505459338003}{6198313388500875118451} a + \frac{2113759300122427494607}{6198313388500875118451}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{6706554674420}{403693694464451} a^{19} - \frac{89383844648180}{403693694464451} a^{18} + \frac{541014939182380}{403693694464451} a^{17} - \frac{2020612896444575}{403693694464451} a^{16} + \frac{5224877463120845}{403693694464451} a^{15} - \frac{10078311200517830}{403693694464451} a^{14} + \frac{14915272752210910}{403693694464451} a^{13} - \frac{15989726397014435}{403693694464451} a^{12} + \frac{9476406897415675}{403693694464451} a^{11} + \frac{3350912431020886}{403693694464451} a^{10} - \frac{12686586064268240}{403693694464451} a^{9} + \frac{9258345661575060}{403693694464451} a^{8} + \frac{1093139212929785}{403693694464451} a^{7} - \frac{4477063184861365}{403693694464451} a^{6} + \frac{263613033683428}{403693694464451} a^{5} + \frac{62578814797535}{403693694464451} a^{4} + \frac{904346815870245}{403693694464451} a^{3} - \frac{59428567328745}{403693694464451} a^{2} + \frac{107427216452000}{403693694464451} a + \frac{159866328305353}{403693694464451} \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 505062.639199 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T138:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 28 conjugacy class representatives for t20n138
Character table for t20n138 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.1.250000.1, 10.0.250000000000.1

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 $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$2$2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.5.6.2$x^{5} + 15 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
5.5.6.2$x^{5} + 15 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
5.10.13.1$x^{10} + 15 x^{4} + 5$$10$$1$$13$$D_5$$[3/2]_{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$