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Properties

Label 20.6.59313783170...0256.2
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{20}\cdot 3^{19}\cdot 13^{5}\cdot 107^{4}$
Root discriminant $27.46$
Ramified primes $2, 3, 13, 107$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1023

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-23, -166, -119, 850, 230, -1490, 374, 1036, -838, -104, 509, -254, -121, 118, -52, 40, -22, -2, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 10*x^18 - 2*x^17 - 22*x^16 + 40*x^15 - 52*x^14 + 118*x^13 - 121*x^12 - 254*x^11 + 509*x^10 - 104*x^9 - 838*x^8 + 1036*x^7 + 374*x^6 - 1490*x^5 + 230*x^4 + 850*x^3 - 119*x^2 - 166*x - 23)
 
gp: K = bnfinit(x^20 - 4*x^19 + 10*x^18 - 2*x^17 - 22*x^16 + 40*x^15 - 52*x^14 + 118*x^13 - 121*x^12 - 254*x^11 + 509*x^10 - 104*x^9 - 838*x^8 + 1036*x^7 + 374*x^6 - 1490*x^5 + 230*x^4 + 850*x^3 - 119*x^2 - 166*x - 23, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 10 x^{18} - 2 x^{17} - 22 x^{16} + 40 x^{15} - 52 x^{14} + 118 x^{13} - 121 x^{12} - 254 x^{11} + 509 x^{10} - 104 x^{9} - 838 x^{8} + 1036 x^{7} + 374 x^{6} - 1490 x^{5} + 230 x^{4} + 850 x^{3} - 119 x^{2} - 166 x - 23 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-59313783170950766384309600256=-\,2^{20}\cdot 3^{19}\cdot 13^{5}\cdot 107^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.46$
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:  $2, 3, 13, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{13} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{16} + \frac{1}{9} a^{15} - \frac{1}{9} a^{14} + \frac{4}{9} a^{13} - \frac{4}{9} a^{12} + \frac{1}{9} a^{11} - \frac{4}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{4}{9} a^{6} - \frac{2}{9} a^{5} + \frac{2}{9} a^{4} + \frac{1}{9} a^{3} + \frac{2}{9} a^{2} - \frac{2}{9} a - \frac{4}{9}$, $\frac{1}{171} a^{18} + \frac{2}{171} a^{17} + \frac{4}{171} a^{16} + \frac{5}{171} a^{15} - \frac{26}{171} a^{14} + \frac{41}{171} a^{13} + \frac{73}{171} a^{12} + \frac{8}{171} a^{11} + \frac{4}{9} a^{10} + \frac{77}{171} a^{9} - \frac{74}{171} a^{8} + \frac{23}{171} a^{7} + \frac{43}{171} a^{6} - \frac{13}{171} a^{5} - \frac{41}{171} a^{4} - \frac{1}{171} a^{3} + \frac{31}{171} a^{2} - \frac{67}{171} a - \frac{8}{19}$, $\frac{1}{3453527882227599378470385} a^{19} - \frac{220920131093874696844}{76745064049502208410453} a^{18} + \frac{3956278881636102891156}{76745064049502208410453} a^{17} - \frac{7946761881159947895476}{60588208460133322429305} a^{16} + \frac{7486280471448708474978}{76745064049502208410453} a^{15} + \frac{8293533702601452990443}{230235192148506625231359} a^{14} - \frac{150228460715428699317079}{1151175960742533126156795} a^{13} - \frac{83093608327402779383861}{230235192148506625231359} a^{12} - \frac{28382379702601157645929}{88551996980194855858215} a^{11} - \frac{234119468260766020283141}{1151175960742533126156795} a^{10} + \frac{50946556776965300721734}{1151175960742533126156795} a^{9} + \frac{35454841562966056730856}{383725320247511042052265} a^{8} + \frac{18858674790874774500889}{60588208460133322429305} a^{7} - \frac{138215878240213069298208}{383725320247511042052265} a^{6} - \frac{226095194345829372180983}{1151175960742533126156795} a^{5} - \frac{279016759863724976909242}{1151175960742533126156795} a^{4} - \frac{436073249976500077328848}{1151175960742533126156795} a^{3} + \frac{363773776984653729358568}{1151175960742533126156795} a^{2} - \frac{361128299540210580706483}{3453527882227599378470385} a + \frac{179937175366176314338943}{383725320247511042052265}$

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:  $12$
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:  \( 14118831.6409 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 10.6.38998285028352.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 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 R R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ R $20$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$3$3.6.6.7$x^{6} + 3 x + 6$$6$$1$$6$$C_3^2:D_4$$[5/4, 5/4]_{4}^{2}$
3.6.6.7$x^{6} + 3 x + 6$$6$$1$$6$$C_3^2:D_4$$[5/4, 5/4]_{4}^{2}$
3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.5.6$x^{6} + 416$$6$$1$$5$$C_6$$[\ ]_{6}$
107Data not computed