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Properties

Label 20.0.57722337794...4089.3
Degree $20$
Signature $[0, 10]$
Discriminant $11^{16}\cdot 23^{4}\cdot 67^{2}$
Root discriminant $19.41$
Ramified primes $11, 23, 67$
Class number $2$
Class group $[2]$
Galois group 20T263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 21, -9, 64, -31, 101, -123, 136, -95, 88, -45, 3, 32, -36, -2, 27, -29, 17, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 17*x^18 - 29*x^17 + 27*x^16 - 2*x^15 - 36*x^14 + 32*x^13 + 3*x^12 - 45*x^11 + 88*x^10 - 95*x^9 + 136*x^8 - 123*x^7 + 101*x^6 - 31*x^5 + 64*x^4 - 9*x^3 + 21*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 17*x^18 - 29*x^17 + 27*x^16 - 2*x^15 - 36*x^14 + 32*x^13 + 3*x^12 - 45*x^11 + 88*x^10 - 95*x^9 + 136*x^8 - 123*x^7 + 101*x^6 - 31*x^5 + 64*x^4 - 9*x^3 + 21*x^2 - 7*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 17 x^{18} - 29 x^{17} + 27 x^{16} - 2 x^{15} - 36 x^{14} + 32 x^{13} + 3 x^{12} - 45 x^{11} + 88 x^{10} - 95 x^{9} + 136 x^{8} - 123 x^{7} + 101 x^{6} - 31 x^{5} + 64 x^{4} - 9 x^{3} + 21 x^{2} - 7 x + 1 \)

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:  \(57722337794481266110634089=11^{16}\cdot 23^{4}\cdot 67^{2}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{195557867876436966253} a^{19} - \frac{31702126794818664310}{195557867876436966253} a^{18} + \frac{57964625402324848448}{195557867876436966253} a^{17} - \frac{89888264581343910512}{195557867876436966253} a^{16} + \frac{5479900075891039025}{195557867876436966253} a^{15} - \frac{3254146714001206436}{195557867876436966253} a^{14} + \frac{45412153885565139839}{195557867876436966253} a^{13} + \frac{12100726910096067384}{195557867876436966253} a^{12} - \frac{85113916429977647646}{195557867876436966253} a^{11} - \frac{20869870212585282244}{195557867876436966253} a^{10} - \frac{36843286635426155080}{195557867876436966253} a^{9} - \frac{97619173064186100964}{195557867876436966253} a^{8} + \frac{35421948540743869684}{195557867876436966253} a^{7} - \frac{87308910161226935826}{195557867876436966253} a^{6} + \frac{39108409941391414244}{195557867876436966253} a^{5} - \frac{53016481317387073369}{195557867876436966253} a^{4} - \frac{97625434449663831497}{195557867876436966253} a^{3} + \frac{12797512714302145460}{195557867876436966253} a^{2} + \frac{88452204117544873307}{195557867876436966253} a + \frac{17353305204657870113}{195557867876436966253}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 22295.2556348 \)
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

\(\Q(\zeta_{11})^+\), 10.2.330327035621.1, 10.6.330327035621.1, 10.2.113395848049.1

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 ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$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}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
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$$[\ ]^{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$