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Properties

Label 20.0.57722337794...4089.4
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, 3, 21, -83, 194, -315, 356, -224, 136, -49, 114, -161, 58, -24, 49, -48, 10, -1, 6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 6*x^18 - x^17 + 10*x^16 - 48*x^15 + 49*x^14 - 24*x^13 + 58*x^12 - 161*x^11 + 114*x^10 - 49*x^9 + 136*x^8 - 224*x^7 + 356*x^6 - 315*x^5 + 194*x^4 - 83*x^3 + 21*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 6*x^18 - x^17 + 10*x^16 - 48*x^15 + 49*x^14 - 24*x^13 + 58*x^12 - 161*x^11 + 114*x^10 - 49*x^9 + 136*x^8 - 224*x^7 + 356*x^6 - 315*x^5 + 194*x^4 - 83*x^3 + 21*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 6 x^{18} - x^{17} + 10 x^{16} - 48 x^{15} + 49 x^{14} - 24 x^{13} + 58 x^{12} - 161 x^{11} + 114 x^{10} - 49 x^{9} + 136 x^{8} - 224 x^{7} + 356 x^{6} - 315 x^{5} + 194 x^{4} - 83 x^{3} + 21 x^{2} + 3 x + 1 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23} a^{17} + \frac{8}{23} a^{16} + \frac{5}{23} a^{15} - \frac{7}{23} a^{14} + \frac{5}{23} a^{13} - \frac{1}{23} a^{12} - \frac{8}{23} a^{11} - \frac{3}{23} a^{10} - \frac{10}{23} a^{9} + \frac{5}{23} a^{8} + \frac{3}{23} a^{7} + \frac{10}{23} a^{6} + \frac{4}{23} a^{5} - \frac{9}{23} a^{4} - \frac{10}{23} a^{2} - \frac{1}{23} a - \frac{6}{23}$, $\frac{1}{989} a^{18} + \frac{10}{989} a^{17} - \frac{140}{989} a^{16} + \frac{371}{989} a^{15} - \frac{446}{989} a^{14} - \frac{428}{989} a^{13} + \frac{82}{989} a^{12} + \frac{280}{989} a^{11} + \frac{421}{989} a^{10} - \frac{130}{989} a^{9} + \frac{128}{989} a^{8} - \frac{191}{989} a^{7} + \frac{484}{989} a^{6} + \frac{160}{989} a^{5} + \frac{74}{989} a^{4} + \frac{174}{989} a^{3} + \frac{278}{989} a^{2} - \frac{31}{989} a + \frac{494}{989}$, $\frac{1}{66557829497876713019} a^{19} + \frac{19698548288704715}{66557829497876713019} a^{18} - \frac{330160457085715248}{66557829497876713019} a^{17} + \frac{14888715872105815010}{66557829497876713019} a^{16} - \frac{1064752199471433724}{2893818673820726653} a^{15} - \frac{19440386675798881146}{66557829497876713019} a^{14} + \frac{26384297925978670769}{66557829497876713019} a^{13} + \frac{16180746788766284935}{66557829497876713019} a^{12} - \frac{6232767323794910581}{66557829497876713019} a^{11} + \frac{15289982788387431796}{66557829497876713019} a^{10} - \frac{770158514667480986}{66557829497876713019} a^{9} + \frac{8366108236876821793}{66557829497876713019} a^{8} - \frac{25706550534937424726}{66557829497876713019} a^{7} - \frac{13809594414319617242}{66557829497876713019} a^{6} - \frac{17969806323719506616}{66557829497876713019} a^{5} - \frac{27844238130974898179}{66557829497876713019} a^{4} - \frac{33046255026563661685}{66557829497876713019} a^{3} - \frac{20506627771909415935}{66557829497876713019} a^{2} + \frac{31376415779617129108}{66557829497876713019} a + \frac{26266972702885738483}{66557829497876713019}$

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:  \( 19549.5881568 \)
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

\(\Q(\zeta_{11})^+\), 10.4.7597521819283.1, 10.4.14362045027.1, 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$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$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{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}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$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$$[\ ]_{2}$
$67$$\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$Trivial$[\ ]$
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}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$