↑

Properties

Label 20.20.5957200041...6416.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{24}\cdot 1429^{10}$
Root discriminant $86.85$
Ramified primes $2, 1429$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^4:D_5$ (as 20T45)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3655744, 0, -12777520, 0, 18445640, 0, -14361184, 0, 6629588, 0, -1886957, 0, 334817, 0, -36486, 0, 2322, 0, -77, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 77*x^18 + 2322*x^16 - 36486*x^14 + 334817*x^12 - 1886957*x^10 + 6629588*x^8 - 14361184*x^6 + 18445640*x^4 - 12777520*x^2 + 3655744)
 
gp: K = bnfinit(x^20 - 77*x^18 + 2322*x^16 - 36486*x^14 + 334817*x^12 - 1886957*x^10 + 6629588*x^8 - 14361184*x^6 + 18445640*x^4 - 12777520*x^2 + 3655744, 1)
 

Normalized defining polynomial

\( x^{20} - 77 x^{18} + 2322 x^{16} - 36486 x^{14} + 334817 x^{12} - 1886957 x^{10} + 6629588 x^{8} - 14361184 x^{6} + 18445640 x^{4} - 12777520 x^{2} + 3655744 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(595720004130752289758344601308198076416=2^{24}\cdot 1429^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.85$
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:  $2, 1429$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{1}{8} a^{6}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{12} - \frac{1}{16} a^{10} + \frac{1}{16} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{13} - \frac{1}{16} a^{11} + \frac{1}{16} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{10} - \frac{1}{16} a^{8} + \frac{1}{8} a^{6}$, $\frac{1}{32} a^{17} - \frac{1}{16} a^{13} - \frac{1}{8} a^{10} + \frac{1}{32} a^{9} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{247683565506665504} a^{18} + \frac{2145129222560739}{123841782753332752} a^{16} - \frac{3604094003028883}{123841782753332752} a^{14} + \frac{226557899747972}{7740111422083297} a^{12} + \frac{2981479608110241}{247683565506665504} a^{10} + \frac{1717227574303345}{123841782753332752} a^{8} + \frac{1939984683882217}{61920891376666376} a^{6} - \frac{987579424754109}{30960445688333188} a^{4} - \frac{1}{2} a^{3} + \frac{4073086127453473}{15480222844166594} a^{2} - \frac{2219293557675954}{7740111422083297}$, $\frac{1}{118392744312186110912} a^{19} - \frac{1303788571886955715}{118392744312186110912} a^{17} - \frac{1}{32} a^{16} + \frac{1745661087387796239}{59196372156093055456} a^{15} + \frac{2689443589858871611}{59196372156093055456} a^{13} - \frac{1}{16} a^{12} + \frac{6690437748288078849}{118392744312186110912} a^{11} + \frac{5785297687444829549}{118392744312186110912} a^{9} + \frac{3}{32} a^{8} - \frac{1961148253156174681}{14799093039023263864} a^{7} - \frac{1}{4} a^{6} - \frac{451952451692206726}{1849886629877907983} a^{5} - \frac{1625017337804668721}{14799093039023263864} a^{3} - \frac{1}{4} a^{2} + \frac{1953809602671722233}{7399546519511631932} a - \frac{1}{2}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $19$
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:  \( 38855057911700 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:D_5$ (as 20T45):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 10 conjugacy class representatives for $C_2^4:D_5$
Character table for $C_2^4:D_5$

Intermediate fields

5.5.2042041.1, 10.10.266875612523584.1, 10.10.24407376018956898304.1, 10.10.381365250296201536.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
1429Data not computed