↑

Properties

Label 20.10.1215000000...0000.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{30}\cdot 3^{5}\cdot 5^{31}$
Root discriminant $45.11$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times D_5\wr C_2$ (as 20T92)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -60, 860, 930, 250, -2388, -14565, -22560, -24005, -13500, 15884, 24330, 3820, -8190, -3375, 888, 580, -30, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 40*x^18 - 30*x^17 + 580*x^16 + 888*x^15 - 3375*x^14 - 8190*x^13 + 3820*x^12 + 24330*x^11 + 15884*x^10 - 13500*x^9 - 24005*x^8 - 22560*x^7 - 14565*x^6 - 2388*x^5 + 250*x^4 + 930*x^3 + 860*x^2 - 60*x + 1)
 
gp: K = bnfinit(x^20 - 40*x^18 - 30*x^17 + 580*x^16 + 888*x^15 - 3375*x^14 - 8190*x^13 + 3820*x^12 + 24330*x^11 + 15884*x^10 - 13500*x^9 - 24005*x^8 - 22560*x^7 - 14565*x^6 - 2388*x^5 + 250*x^4 + 930*x^3 + 860*x^2 - 60*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 40 x^{18} - 30 x^{17} + 580 x^{16} + 888 x^{15} - 3375 x^{14} - 8190 x^{13} + 3820 x^{12} + 24330 x^{11} + 15884 x^{10} - 13500 x^{9} - 24005 x^{8} - 22560 x^{7} - 14565 x^{6} - 2388 x^{5} + 250 x^{4} + 930 x^{3} + 860 x^{2} - 60 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1215000000000000000000000000000000=-\,2^{30}\cdot 3^{5}\cdot 5^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.11$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{5} - \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{6} - \frac{1}{5} a$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{7} - \frac{1}{5} a^{2}$, $\frac{1}{75} a^{13} + \frac{2}{25} a^{12} + \frac{2}{75} a^{11} - \frac{2}{75} a^{10} + \frac{2}{15} a^{9} + \frac{19}{75} a^{8} + \frac{8}{25} a^{7} - \frac{9}{25} a^{6} + \frac{32}{75} a^{5} - \frac{2}{5} a^{4} - \frac{26}{75} a^{3} + \frac{14}{75} a^{2} - \frac{7}{75} a + \frac{17}{75}$, $\frac{1}{75} a^{14} - \frac{4}{75} a^{12} + \frac{1}{75} a^{11} + \frac{7}{75} a^{10} + \frac{34}{75} a^{9} - \frac{1}{5} a^{8} + \frac{8}{25} a^{7} + \frac{29}{75} a^{6} + \frac{6}{25} a^{5} + \frac{4}{75} a^{4} + \frac{4}{15} a^{3} + \frac{29}{75} a^{2} - \frac{31}{75} a - \frac{4}{25}$, $\frac{1}{75} a^{15} - \frac{1}{15} a^{12} - \frac{4}{75} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{15} a^{7} + \frac{4}{25} a^{5} - \frac{1}{3} a^{4} - \frac{4}{15} a^{2} - \frac{1}{3} a + \frac{23}{75}$, $\frac{1}{75} a^{16} + \frac{2}{25} a^{11} + \frac{1}{3} a^{8} + \frac{9}{25} a^{6} - \frac{4}{25} a + \frac{1}{3}$, $\frac{1}{75} a^{17} + \frac{2}{25} a^{12} + \frac{1}{3} a^{9} + \frac{9}{25} a^{7} - \frac{4}{25} a^{2} + \frac{1}{3} a$, $\frac{1}{225} a^{18} + \frac{1}{225} a^{16} - \frac{2}{75} a^{12} + \frac{1}{25} a^{11} - \frac{8}{225} a^{10} + \frac{1}{15} a^{9} + \frac{13}{225} a^{8} - \frac{8}{75} a^{7} - \frac{17}{75} a^{6} + \frac{26}{75} a^{5} - \frac{1}{5} a^{4} - \frac{9}{25} a^{3} + \frac{61}{225} a^{2} + \frac{2}{5} a - \frac{32}{225}$, $\frac{1}{158242743075077324066925} a^{19} - \frac{15736094649885137609}{52747581025025774688975} a^{18} - \frac{510935037826771922987}{158242743075077324066925} a^{17} + \frac{2351816645783305346}{17582527008341924896325} a^{16} - \frac{58714140032167715888}{17582527008341924896325} a^{15} + \frac{8589475098292490983}{10549516205005154937795} a^{14} - \frac{2798148735455914}{1614310054323665637} a^{13} + \frac{2207181086494959314548}{52747581025025774688975} a^{12} + \frac{12070921508433066896467}{158242743075077324066925} a^{11} - \frac{1129389165126030804706}{52747581025025774688975} a^{10} - \frac{36702672850574477300927}{158242743075077324066925} a^{9} + \frac{17182544615927169423538}{52747581025025774688975} a^{8} - \frac{268485351818777330105}{2109903241001030987559} a^{7} - \frac{780558921220471194308}{3516505401668384979265} a^{6} - \frac{3424893408151823345612}{17582527008341924896325} a^{5} + \frac{11216751749996249854403}{52747581025025774688975} a^{4} + \frac{55470094364253339821242}{158242743075077324066925} a^{3} + \frac{17034838409120806607156}{52747581025025774688975} a^{2} - \frac{31901792214760175794406}{158242743075077324066925} a + \frac{1205009961965628893844}{3516505401668384979265}$

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:  $14$
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:  \( 4882656633.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_5\wr C_2$ (as 20T92):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $C_2\times D_5\wr C_2$
Character table for $C_2\times D_5\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), 4.2.24000.1, 10.6.9000000000000000.1

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5Data not computed