↑

Properties

Label 20.12.3491991015...6481.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{4}\cdot 401^{11}$
Root discriminant $33.66$
Ramified primes $3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times C_2^4:C_5).C_2$ (as 20T84)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-169, 338, 24, -347, 1674, -4549, 788, 9999, -7280, -6886, 7201, 341, -2910, 1142, 443, -371, 34, 45, -15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 15*x^18 + 45*x^17 + 34*x^16 - 371*x^15 + 443*x^14 + 1142*x^13 - 2910*x^12 + 341*x^11 + 7201*x^10 - 6886*x^9 - 7280*x^8 + 9999*x^7 + 788*x^6 - 4549*x^5 + 1674*x^4 - 347*x^3 + 24*x^2 + 338*x - 169)
 
gp: K = bnfinit(x^20 - 2*x^19 - 15*x^18 + 45*x^17 + 34*x^16 - 371*x^15 + 443*x^14 + 1142*x^13 - 2910*x^12 + 341*x^11 + 7201*x^10 - 6886*x^9 - 7280*x^8 + 9999*x^7 + 788*x^6 - 4549*x^5 + 1674*x^4 - 347*x^3 + 24*x^2 + 338*x - 169, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 15 x^{18} + 45 x^{17} + 34 x^{16} - 371 x^{15} + 443 x^{14} + 1142 x^{13} - 2910 x^{12} + 341 x^{11} + 7201 x^{10} - 6886 x^{9} - 7280 x^{8} + 9999 x^{7} + 788 x^{6} - 4549 x^{5} + 1674 x^{4} - 347 x^{3} + 24 x^{2} + 338 x - 169 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{14} + \frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{4}{9} a^{10} - \frac{2}{9} a^{9} + \frac{1}{3} a^{8} + \frac{2}{9} a^{7} + \frac{1}{9} a^{5} - \frac{1}{3} a^{4} + \frac{4}{9} a^{3} + \frac{4}{9} a^{2} - \frac{2}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{15} + \frac{1}{9} a^{14} - \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{2}{9} a^{10} + \frac{1}{3} a^{9} - \frac{4}{9} a^{8} - \frac{1}{3} a^{7} - \frac{2}{9} a^{6} + \frac{1}{3} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{3} + \frac{4}{9} a^{2} + \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{2223} a^{18} + \frac{112}{2223} a^{17} + \frac{1}{171} a^{16} + \frac{28}{741} a^{15} + \frac{94}{2223} a^{14} + \frac{62}{2223} a^{13} - \frac{1}{741} a^{12} + \frac{85}{2223} a^{11} - \frac{314}{741} a^{10} + \frac{346}{2223} a^{9} + \frac{794}{2223} a^{8} - \frac{181}{2223} a^{7} + \frac{322}{2223} a^{6} + \frac{73}{741} a^{5} + \frac{50}{117} a^{4} + \frac{1051}{2223} a^{3} + \frac{952}{2223} a^{2} + \frac{124}{741} a - \frac{64}{171}$, $\frac{1}{2711074087276070756589999} a^{19} - \frac{147180027642868700885}{903691362425356918863333} a^{18} - \frac{27238892659955201316293}{903691362425356918863333} a^{17} - \frac{3406002578431046315937}{100410151380595213207037} a^{16} + \frac{72617583903203309886445}{2711074087276070756589999} a^{15} - \frac{668029488579047089201}{301230454141785639621111} a^{14} - \frac{6168285362789267019610}{208544160559697750506923} a^{13} - \frac{9658160779680540933640}{100410151380595213207037} a^{12} - \frac{8045339995522987097282}{69514720186565916835641} a^{11} - \frac{313388316169010536993837}{2711074087276070756589999} a^{10} - \frac{1053811251740945936631946}{2711074087276070756589999} a^{9} - \frac{28279006816604536918976}{69514720186565916835641} a^{8} + \frac{1206715532082479688740584}{2711074087276070756589999} a^{7} - \frac{1060130955549328149355114}{2711074087276070756589999} a^{6} + \frac{358368337504049653553189}{903691362425356918863333} a^{5} - \frac{361643768833629734475793}{2711074087276070756589999} a^{4} - \frac{1323993128024699779097576}{2711074087276070756589999} a^{3} - \frac{56630650197708664228435}{301230454141785639621111} a^{2} - \frac{49772859036298927734056}{903691362425356918863333} a + \frac{82828766559478276139984}{208544160559697750506923}$

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:  $15$
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:  \( 145604533.377 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_2^4:C_5).C_2$ (as 20T84):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $(C_2\times C_2^4:C_5).C_2$
Character table for $(C_2\times C_2^4:C_5).C_2$

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

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}$
3.4.2.1$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}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed