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Properties

Label 20.0.68020707707...0409.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 181^{8}$
Root discriminant $13.86$
Ramified primes $3, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T226

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 3, 0, -2, -1, -6, 4, -1, -1, 11, -1, -1, 4, -6, -1, -2, 0, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 3*x^18 - 2*x^16 - x^15 - 6*x^14 + 4*x^13 - x^12 - x^11 + 11*x^10 - x^9 - x^8 + 4*x^7 - 6*x^6 - x^5 - 2*x^4 + 3*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 3*x^18 - 2*x^16 - x^15 - 6*x^14 + 4*x^13 - x^12 - x^11 + 11*x^10 - x^9 - x^8 + 4*x^7 - 6*x^6 - x^5 - 2*x^4 + 3*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 3 x^{18} - 2 x^{16} - x^{15} - 6 x^{14} + 4 x^{13} - x^{12} - x^{11} + 11 x^{10} - x^{9} - x^{8} + 4 x^{7} - 6 x^{6} - x^{5} - 2 x^{4} + 3 x^{2} - 2 x + 1 \)

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:  \(68020707707819889350409=3^{10}\cdot 181^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.86$
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:  $3, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3}$, $\frac{1}{7368} a^{18} + \frac{1049}{7368} a^{17} - \frac{81}{2456} a^{16} - \frac{579}{1228} a^{15} - \frac{439}{2456} a^{14} - \frac{211}{3684} a^{13} + \frac{775}{2456} a^{12} + \frac{2737}{7368} a^{11} + \frac{247}{2456} a^{10} - \frac{43}{7368} a^{9} + \frac{247}{2456} a^{8} + \frac{2737}{7368} a^{7} + \frac{775}{2456} a^{6} - \frac{211}{3684} a^{5} - \frac{439}{2456} a^{4} - \frac{579}{1228} a^{3} - \frac{81}{2456} a^{2} + \frac{1049}{7368} a + \frac{1}{7368}$, $\frac{1}{7368} a^{19} - \frac{89}{1842} a^{17} + \frac{1}{8} a^{16} + \frac{1039}{2456} a^{15} + \frac{3295}{7368} a^{14} + \frac{2923}{7368} a^{13} - \frac{191}{614} a^{12} - \frac{147}{614} a^{11} + \frac{50}{307} a^{10} - \frac{34}{307} a^{9} + \frac{127}{614} a^{8} - \frac{15}{614} a^{7} - \frac{2995}{7368} a^{6} - \frac{3175}{7368} a^{5} + \frac{81}{2456} a^{4} - \frac{1059}{2456} a^{3} - \frac{481}{1842} a^{2} - \frac{107}{307} a - \frac{3505}{7368}$

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:  \( \frac{349}{307} a^{19} - \frac{1358}{921} a^{18} + \frac{2053}{921} a^{17} + \frac{1811}{921} a^{16} - \frac{1423}{921} a^{15} - \frac{1525}{921} a^{14} - \frac{2521}{307} a^{13} - \frac{374}{921} a^{12} - \frac{914}{921} a^{11} - \frac{2846}{921} a^{10} + \frac{9610}{921} a^{9} + \frac{4822}{921} a^{8} + \frac{1576}{921} a^{7} + \frac{5911}{921} a^{6} - \frac{757}{307} a^{5} - \frac{1099}{921} a^{4} - \frac{2407}{921} a^{3} - \frac{2074}{921} a^{2} + \frac{2404}{921} a - \frac{908}{921} \) (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:  \( 5277.24530909 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T226:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1920
The 18 conjugacy class representatives for t20n226
Character table for t20n226

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.3.98283.1, 10.0.260807798403.2, 10.2.9659548089.1, 10.4.260807798403.1

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 30 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$3$3.4.2.1$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}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.6.4.3$x^{6} + 6335 x^{3} + 10614564$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
181.6.4.3$x^{6} + 6335 x^{3} + 10614564$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$