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Properties

Label 20.0.15583578925...2688.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{5}\cdot 11^{19}$
Root discriminant $25.68$
Ramified primes $2, 3, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T427

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 38, 346, -1956, 4409, -5858, 5675, -5176, 5094, -4660, 3553, -2450, 1720, -1074, 435, -72, 15, -36, 26, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 26*x^18 - 36*x^17 + 15*x^16 - 72*x^15 + 435*x^14 - 1074*x^13 + 1720*x^12 - 2450*x^11 + 3553*x^10 - 4660*x^9 + 5094*x^8 - 5176*x^7 + 5675*x^6 - 5858*x^5 + 4409*x^4 - 1956*x^3 + 346*x^2 + 38*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 26*x^18 - 36*x^17 + 15*x^16 - 72*x^15 + 435*x^14 - 1074*x^13 + 1720*x^12 - 2450*x^11 + 3553*x^10 - 4660*x^9 + 5094*x^8 - 5176*x^7 + 5675*x^6 - 5858*x^5 + 4409*x^4 - 1956*x^3 + 346*x^2 + 38*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 26 x^{18} - 36 x^{17} + 15 x^{16} - 72 x^{15} + 435 x^{14} - 1074 x^{13} + 1720 x^{12} - 2450 x^{11} + 3553 x^{10} - 4660 x^{9} + 5094 x^{8} - 5176 x^{7} + 5675 x^{6} - 5858 x^{5} + 4409 x^{4} - 1956 x^{3} + 346 x^{2} + 38 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15583578925526925703866482688=2^{20}\cdot 3^{5}\cdot 11^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.68$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1113282657326395273178947531} a^{19} + \frac{385821430073340384152199578}{1113282657326395273178947531} a^{18} + \frac{375672666998675925222884132}{1113282657326395273178947531} a^{17} + \frac{123670245330516776189638070}{1113282657326395273178947531} a^{16} - \frac{241390636130644621175311662}{1113282657326395273178947531} a^{15} - \frac{207769798790680098719409946}{1113282657326395273178947531} a^{14} - \frac{237685400456947797858503120}{1113282657326395273178947531} a^{13} + \frac{3102006572819163130701772}{10213602360792617185127959} a^{12} - \frac{223880651126306235497927729}{1113282657326395273178947531} a^{11} - \frac{423819673856629678930792480}{1113282657326395273178947531} a^{10} + \frac{282745217520299633493439701}{1113282657326395273178947531} a^{9} - \frac{95764486275488300896501028}{1113282657326395273178947531} a^{8} - \frac{16890636184993780366833328}{48403593796799794486041197} a^{7} + \frac{315630025112178413613730063}{1113282657326395273178947531} a^{6} - \frac{468462446248980879666568485}{1113282657326395273178947531} a^{5} + \frac{547674444980868459000326380}{1113282657326395273178947531} a^{4} - \frac{10495007229513801989722102}{1113282657326395273178947531} a^{3} - \frac{351672036156013205022247093}{1113282657326395273178947531} a^{2} + \frac{367579493533819359366067}{10213602360792617185127959} a + \frac{474876314463184202836910888}{1113282657326395273178947531}$

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:  $9$
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:  \( 338990.621379 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T427:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n427 are not computed
Character table for t20n427 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.2414538435584.2

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 ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ $20$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$2$2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
$3$3.10.0.1$x^{10} - x^{3} - x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed