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Properties

Label 20.8.34919910159...6481.5
Degree $20$
Signature $[8, 6]$
Discriminant $3^{4}\cdot 401^{11}$
Root discriminant $33.66$
Ramified primes $3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T350

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 108, 219, -468, -1324, 607, 1665, -426, 1269, -1140, -387, 766, -650, 266, 7, -66, 76, -39, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 10*x^18 - 39*x^17 + 76*x^16 - 66*x^15 + 7*x^14 + 266*x^13 - 650*x^12 + 766*x^11 - 387*x^10 - 1140*x^9 + 1269*x^8 - 426*x^7 + 1665*x^6 + 607*x^5 - 1324*x^4 - 468*x^3 + 219*x^2 + 108*x + 9)
 
gp: K = bnfinit(x^20 - 4*x^19 + 10*x^18 - 39*x^17 + 76*x^16 - 66*x^15 + 7*x^14 + 266*x^13 - 650*x^12 + 766*x^11 - 387*x^10 - 1140*x^9 + 1269*x^8 - 426*x^7 + 1665*x^6 + 607*x^5 - 1324*x^4 - 468*x^3 + 219*x^2 + 108*x + 9, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 10 x^{18} - 39 x^{17} + 76 x^{16} - 66 x^{15} + 7 x^{14} + 266 x^{13} - 650 x^{12} + 766 x^{11} - 387 x^{10} - 1140 x^{9} + 1269 x^{8} - 426 x^{7} + 1665 x^{6} + 607 x^{5} - 1324 x^{4} - 468 x^{3} + 219 x^{2} + 108 x + 9 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3491991015954496398424073156481=3^{4}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.66$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{39} a^{18} + \frac{1}{13} a^{17} + \frac{4}{39} a^{16} - \frac{1}{39} a^{15} + \frac{7}{39} a^{12} + \frac{1}{13} a^{11} - \frac{4}{13} a^{10} + \frac{16}{39} a^{9} - \frac{16}{39} a^{8} - \frac{7}{39} a^{7} - \frac{4}{13} a^{6} + \frac{4}{39} a^{5} + \frac{2}{39} a^{4} + \frac{19}{39} a^{3} - \frac{10}{39} a^{2} + \frac{5}{13} a + \frac{3}{13}$, $\frac{1}{9784006881099716867923118811} a^{19} + \frac{10136981107862410218734146}{3261335627033238955974372937} a^{18} + \frac{974170610535581466437883001}{9784006881099716867923118811} a^{17} - \frac{920453997214377011586986017}{9784006881099716867923118811} a^{16} - \frac{123659760819078315583150013}{9784006881099716867923118811} a^{15} - \frac{1535462635993203127044307}{11233073342249961960876141} a^{14} + \frac{3932424818773221982722556910}{9784006881099716867923118811} a^{13} - \frac{4185961152131877746905209928}{9784006881099716867923118811} a^{12} - \frac{196707465627882124490835308}{752615913930747451378701447} a^{11} + \frac{3613034036543738706619968710}{9784006881099716867923118811} a^{10} + \frac{836017062321514844178792406}{9784006881099716867923118811} a^{9} - \frac{2410654851885804852938591936}{9784006881099716867923118811} a^{8} - \frac{3461090398274038952995725647}{9784006881099716867923118811} a^{7} + \frac{2026508477663928859842912367}{9784006881099716867923118811} a^{6} + \frac{71426658967651432488432520}{171649243528065208209177523} a^{5} + \frac{2830924273962243569226647389}{9784006881099716867923118811} a^{4} - \frac{553541788613614267449399018}{3261335627033238955974372937} a^{3} + \frac{1443565958075683495662750298}{9784006881099716867923118811} a^{2} - \frac{1008201636655461609126857497}{3261335627033238955974372937} a + \frac{1590157860382064077648614615}{3261335627033238955974372937}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 61583452.4265 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T350:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n350 are not computed
Character table for t20n350 is not computed

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.2.0.1$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}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed