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Properties

Label 20.0.41080775966...6441.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 17^{4}\cdot 97^{6}$
Root discriminant $12.04$
Ramified primes $3, 17, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T368

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 1, -1, 6, 1, -20, 1, 63, -68, -17, 59, 9, -78, 74, -53, 54, -49, 27, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 27*x^18 - 49*x^17 + 54*x^16 - 53*x^15 + 74*x^14 - 78*x^13 + 9*x^12 + 59*x^11 - 17*x^10 - 68*x^9 + 63*x^8 + x^7 - 20*x^6 + x^5 + 6*x^4 - x^3 + x^2 - 2*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 27*x^18 - 49*x^17 + 54*x^16 - 53*x^15 + 74*x^14 - 78*x^13 + 9*x^12 + 59*x^11 - 17*x^10 - 68*x^9 + 63*x^8 + x^7 - 20*x^6 + x^5 + 6*x^4 - x^3 + x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 27 x^{18} - 49 x^{17} + 54 x^{16} - 53 x^{15} + 74 x^{14} - 78 x^{13} + 9 x^{12} + 59 x^{11} - 17 x^{10} - 68 x^{9} + 63 x^{8} + x^{7} - 20 x^{6} + x^{5} + 6 x^{4} - x^{3} + x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4108077596683185606441=3^{10}\cdot 17^{4}\cdot 97^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.04$
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:  $3, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11} a^{18} + \frac{1}{11} a^{17} - \frac{3}{11} a^{16} - \frac{5}{11} a^{15} + \frac{5}{11} a^{14} - \frac{1}{11} a^{11} - \frac{1}{11} a^{9} - \frac{4}{11} a^{8} + \frac{1}{11} a^{7} - \frac{3}{11} a^{6} + \frac{1}{11} a^{5} - \frac{4}{11} a^{4} + \frac{3}{11} a^{3} + \frac{2}{11} a^{2} - \frac{1}{11} a + \frac{2}{11}$, $\frac{1}{62953} a^{19} - \frac{1544}{62953} a^{18} - \frac{14880}{62953} a^{17} + \frac{9415}{62953} a^{16} + \frac{635}{62953} a^{15} + \frac{3220}{62953} a^{14} - \frac{627}{5723} a^{13} + \frac{17610}{62953} a^{12} + \frac{20839}{62953} a^{11} + \frac{28709}{62953} a^{10} - \frac{29941}{62953} a^{9} + \frac{27895}{62953} a^{8} + \frac{1444}{62953} a^{7} - \frac{8905}{62953} a^{6} + \frac{90}{62953} a^{5} - \frac{6610}{62953} a^{4} - \frac{5359}{62953} a^{3} + \frac{159}{5723} a^{2} - \frac{30991}{62953} a + \frac{3983}{62953}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{3551}{5723} a^{19} + \frac{17279}{5723} a^{18} - \frac{18748}{5723} a^{17} - \frac{38960}{5723} a^{16} + \frac{91545}{5723} a^{15} - \frac{16835}{5723} a^{14} - \frac{14639}{5723} a^{13} - \frac{186748}{5723} a^{12} + \frac{210852}{5723} a^{11} + \frac{181276}{5723} a^{10} - \frac{310445}{5723} a^{9} - \frac{93029}{5723} a^{8} + \frac{269145}{5723} a^{7} + \frac{13526}{5723} a^{6} - \frac{130731}{5723} a^{5} - \frac{20805}{5723} a^{4} + \frac{29449}{5723} a^{3} + \frac{15925}{5723} a^{2} - \frac{9972}{5723} a - \frac{7823}{5723} \) (order $6$)
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:  \( 654.41934131 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T368:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 72 conjugacy class representatives for t20n368 are not computed
Character table for t20n368 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.1.1649.1, 10.0.660765843.1, 10.0.7121587419.1, 10.2.2373862473.1

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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.3.1$x^{4} - 97$$4$$1$$3$$C_4$$[\ ]_{4}$
97.4.3.1$x^{4} - 97$$4$$1$$3$$C_4$$[\ ]_{4}$