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Properties

Label 20.8.57722337794...4089.3
Degree $20$
Signature $[8, 6]$
Discriminant $11^{16}\cdot 23^{4}\cdot 67^{2}$
Root discriminant $19.41$
Ramified primes $11, 23, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 15, -68, 112, 55, -451, 568, -49, -628, 845, -407, -196, 375, -193, 26, 33, -44, 20, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 2*x^18 + 20*x^17 - 44*x^16 + 33*x^15 + 26*x^14 - 193*x^13 + 375*x^12 - 196*x^11 - 407*x^10 + 845*x^9 - 628*x^8 - 49*x^7 + 568*x^6 - 451*x^5 + 55*x^4 + 112*x^3 - 68*x^2 + 15*x - 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 2*x^18 + 20*x^17 - 44*x^16 + 33*x^15 + 26*x^14 - 193*x^13 + 375*x^12 - 196*x^11 - 407*x^10 + 845*x^9 - 628*x^8 - 49*x^7 + 568*x^6 - 451*x^5 + 55*x^4 + 112*x^3 - 68*x^2 + 15*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 2 x^{18} + 20 x^{17} - 44 x^{16} + 33 x^{15} + 26 x^{14} - 193 x^{13} + 375 x^{12} - 196 x^{11} - 407 x^{10} + 845 x^{9} - 628 x^{8} - 49 x^{7} + 568 x^{6} - 451 x^{5} + 55 x^{4} + 112 x^{3} - 68 x^{2} + 15 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(57722337794481266110634089=11^{16}\cdot 23^{4}\cdot 67^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.41$
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:  $11, 23, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11} a^{18} + \frac{3}{11} a^{17} - \frac{3}{11} a^{16} - \frac{2}{11} a^{15} - \frac{2}{11} a^{14} + \frac{5}{11} a^{13} + \frac{3}{11} a^{12} - \frac{5}{11} a^{11} - \frac{2}{11} a^{10} - \frac{3}{11} a^{9} - \frac{2}{11} a^{8} - \frac{4}{11} a^{7} + \frac{1}{11} a^{6} - \frac{4}{11} a^{5} - \frac{3}{11} a^{4} - \frac{5}{11} a^{3} - \frac{1}{11} a^{2} + \frac{4}{11} a - \frac{3}{11}$, $\frac{1}{125335697550550927} a^{19} + \frac{2065724036335389}{125335697550550927} a^{18} + \frac{1790852391420094}{125335697550550927} a^{17} - \frac{60889113034471498}{125335697550550927} a^{16} + \frac{39571008585559750}{125335697550550927} a^{15} + \frac{25782972643544136}{125335697550550927} a^{14} - \frac{44273051657015199}{125335697550550927} a^{13} - \frac{5101350513253906}{11394154322777357} a^{12} - \frac{53656761742631765}{125335697550550927} a^{11} - \frac{34708739975684708}{125335697550550927} a^{10} + \frac{57677591277930606}{125335697550550927} a^{9} + \frac{3206348787501374}{11394154322777357} a^{8} + \frac{24160880818146334}{125335697550550927} a^{7} - \frac{33735482797973435}{125335697550550927} a^{6} + \frac{43004673751229723}{125335697550550927} a^{5} + \frac{5825450989395027}{125335697550550927} a^{4} - \frac{7375951059600257}{125335697550550927} a^{3} + \frac{54275162045199640}{125335697550550927} a^{2} - \frac{1667515979097907}{11394154322777357} a + \frac{8612631607201331}{125335697550550927}$

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:  \( 212810.98689 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2560
The 112 conjugacy class representatives for t20n263 are not computed
Character table for t20n263 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.330327035621.4, 10.6.330327035621.2, 10.6.113395848049.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$67$67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$