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Properties

Label 20.4.15967166941...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{38}\cdot 5^{10}\cdot 29^{6}$
Root discriminant $22.92$
Ramified primes $2, 5, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T201

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 10, -59, 246, -781, 1964, -3910, 6056, -7257, 6882, -5417, 3650, -1987, 696, -6, -156, 101, -42, 17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 17*x^18 - 42*x^17 + 101*x^16 - 156*x^15 - 6*x^14 + 696*x^13 - 1987*x^12 + 3650*x^11 - 5417*x^10 + 6882*x^9 - 7257*x^8 + 6056*x^7 - 3910*x^6 + 1964*x^5 - 781*x^4 + 246*x^3 - 59*x^2 + 10*x - 1)
 
gp: K = bnfinit(x^20 - 6*x^19 + 17*x^18 - 42*x^17 + 101*x^16 - 156*x^15 - 6*x^14 + 696*x^13 - 1987*x^12 + 3650*x^11 - 5417*x^10 + 6882*x^9 - 7257*x^8 + 6056*x^7 - 3910*x^6 + 1964*x^5 - 781*x^4 + 246*x^3 - 59*x^2 + 10*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 17 x^{18} - 42 x^{17} + 101 x^{16} - 156 x^{15} - 6 x^{14} + 696 x^{13} - 1987 x^{12} + 3650 x^{11} - 5417 x^{10} + 6882 x^{9} - 7257 x^{8} + 6056 x^{7} - 3910 x^{6} + 1964 x^{5} - 781 x^{4} + 246 x^{3} - 59 x^{2} + 10 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1596716694120693760000000000=2^{38}\cdot 5^{10}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.92$
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:  $2, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{13} + \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{8} + \frac{3}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{1}{4} a^{7} + \frac{3}{8} a^{6} - \frac{3}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{8}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{8} a$, $\frac{1}{8152} a^{18} + \frac{141}{4076} a^{17} + \frac{111}{2038} a^{16} - \frac{65}{2038} a^{15} + \frac{65}{4076} a^{14} + \frac{37}{4076} a^{13} + \frac{85}{4076} a^{12} + \frac{925}{4076} a^{11} - \frac{1153}{8152} a^{10} - \frac{589}{4076} a^{9} + \frac{315}{4076} a^{8} + \frac{891}{4076} a^{7} + \frac{1767}{4076} a^{6} - \frac{368}{1019} a^{5} - \frac{1383}{4076} a^{4} - \frac{899}{4076} a^{3} - \frac{1203}{8152} a^{2} + \frac{613}{2038} a - \frac{521}{2038}$, $\frac{1}{432056} a^{19} - \frac{1}{108014} a^{18} - \frac{5821}{432056} a^{17} + \frac{18473}{432056} a^{16} - \frac{31}{1019} a^{15} - \frac{2855}{54007} a^{14} - \frac{112}{1019} a^{13} + \frac{7185}{216028} a^{12} + \frac{7779}{432056} a^{11} - \frac{36453}{216028} a^{10} + \frac{89921}{432056} a^{9} - \frac{20453}{432056} a^{8} + \frac{56717}{216028} a^{7} + \frac{52597}{216028} a^{6} - \frac{45055}{216028} a^{5} + \frac{13533}{216028} a^{4} - \frac{22969}{432056} a^{3} + \frac{87659}{216028} a^{2} - \frac{88899}{432056} a - \frac{53079}{432056}$

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:  $11$
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:  \( 1102675.70668 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T201:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1440
The 13 conjugacy class representatives for t20n201
Character table for t20n201

Intermediate fields

\(\Q(\sqrt{5}) \), 10.2.39958937600000.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 12 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 sibling: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.16.21$x^{8} + 12 x^{7} + 20 x^{5} + 16 x^{4} + 40 x + 20$$4$$2$$16$$Q_8:C_2$$[2, 3, 3]^{2}$
2.8.16.21$x^{8} + 12 x^{7} + 20 x^{5} + 16 x^{4} + 40 x + 20$$4$$2$$16$$Q_8:C_2$$[2, 3, 3]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$