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Properties

Label 20.8.42077520999...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{16}\cdot 3^{2}\cdot 5^{14}\cdot 43^{8}$
Root discriminant $26.99$
Ramified primes $2, 3, 5, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T140

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, 100, -775, 2960, -5989, 6706, -4008, 22, 3781, -6294, 6729, -5744, 4255, -2794, 1604, -820, 383, -156, 49, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 49*x^18 - 156*x^17 + 383*x^16 - 820*x^15 + 1604*x^14 - 2794*x^13 + 4255*x^12 - 5744*x^11 + 6729*x^10 - 6294*x^9 + 3781*x^8 + 22*x^7 - 4008*x^6 + 6706*x^5 - 5989*x^4 + 2960*x^3 - 775*x^2 + 100*x - 5)
 
gp: K = bnfinit(x^20 - 10*x^19 + 49*x^18 - 156*x^17 + 383*x^16 - 820*x^15 + 1604*x^14 - 2794*x^13 + 4255*x^12 - 5744*x^11 + 6729*x^10 - 6294*x^9 + 3781*x^8 + 22*x^7 - 4008*x^6 + 6706*x^5 - 5989*x^4 + 2960*x^3 - 775*x^2 + 100*x - 5, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 49 x^{18} - 156 x^{17} + 383 x^{16} - 820 x^{15} + 1604 x^{14} - 2794 x^{13} + 4255 x^{12} - 5744 x^{11} + 6729 x^{10} - 6294 x^{9} + 3781 x^{8} + 22 x^{7} - 4008 x^{6} + 6706 x^{5} - 5989 x^{4} + 2960 x^{3} - 775 x^{2} + 100 x - 5 \)

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:  \(42077520999363600000000000000=2^{16}\cdot 3^{2}\cdot 5^{14}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.99$
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:  $2, 3, 5, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{13} + \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{2} a^{9} + \frac{1}{8} a^{8} - \frac{3}{8} a^{7} + \frac{1}{8} a^{6} + \frac{1}{4} a^{5} + \frac{3}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{3}{8} a - \frac{3}{8}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{3}{8} a^{6} - \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{1}{8}$, $\frac{1}{16} a^{16} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{8} a^{9} + \frac{7}{16} a^{8} + \frac{3}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} + \frac{5}{16}$, $\frac{1}{16} a^{17} - \frac{1}{4} a^{11} - \frac{1}{8} a^{10} + \frac{7}{16} a^{9} + \frac{3}{8} a^{7} + \frac{3}{8} a^{5} + \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{5}{16} a + \frac{1}{4}$, $\frac{1}{1774048} a^{18} - \frac{9}{1774048} a^{17} + \frac{11645}{1774048} a^{16} - \frac{23239}{443512} a^{15} - \frac{219}{221756} a^{14} - \frac{16971}{221756} a^{13} - \frac{16143}{443512} a^{12} - \frac{150191}{887024} a^{11} - \frac{118907}{1774048} a^{10} + \frac{833583}{1774048} a^{9} + \frac{246897}{1774048} a^{8} - \frac{177969}{887024} a^{7} - \frac{216315}{443512} a^{6} + \frac{7252}{55439} a^{5} + \frac{110237}{443512} a^{4} - \frac{170757}{887024} a^{3} + \frac{56093}{1774048} a^{2} + \frac{455827}{1774048} a + \frac{338701}{1774048}$, $\frac{1}{1875168736} a^{19} + \frac{519}{1875168736} a^{18} - \frac{24386267}{1875168736} a^{17} - \frac{749951}{66970312} a^{16} + \frac{3192346}{58599023} a^{15} + \frac{13838025}{234396092} a^{14} + \frac{1444945}{66970312} a^{13} - \frac{59552595}{937584368} a^{12} - \frac{274033723}{1875168736} a^{11} - \frac{280600729}{1875168736} a^{10} + \frac{223944865}{1875168736} a^{9} + \frac{228880523}{937584368} a^{8} - \frac{116388003}{468792184} a^{7} - \frac{1540941}{33485156} a^{6} - \frac{149212309}{468792184} a^{5} + \frac{408513923}{937584368} a^{4} - \frac{75150955}{1875168736} a^{3} - \frac{28080243}{267881248} a^{2} - \frac{743137771}{1875168736} a - \frac{41379039}{117198046}$

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:  \( 17027908.4731 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T140:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 22 conjugacy class representatives for t20n140
Character table for t20n140 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.3698000.1, 10.4.41025612000000.1, 10.4.205128060000000.1, 10.10.68376020000000.1

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 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$43$43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$