↑

Properties

Label 20.8.15374146899...1728.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{16}\cdot 17\cdot 53^{14}$
Root discriminant $32.31$
Ramified primes $2, 17, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T513

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 13, -26, -251, 442, 261, -394, 1469, -849, 10, 1114, -1484, 1033, -275, -198, 257, -116, 7, 16, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 16*x^18 + 7*x^17 - 116*x^16 + 257*x^15 - 198*x^14 - 275*x^13 + 1033*x^12 - 1484*x^11 + 1114*x^10 + 10*x^9 - 849*x^8 + 1469*x^7 - 394*x^6 + 261*x^5 + 442*x^4 - 251*x^3 - 26*x^2 + 13*x + 1)
 
gp: K = bnfinit(x^20 - 7*x^19 + 16*x^18 + 7*x^17 - 116*x^16 + 257*x^15 - 198*x^14 - 275*x^13 + 1033*x^12 - 1484*x^11 + 1114*x^10 + 10*x^9 - 849*x^8 + 1469*x^7 - 394*x^6 + 261*x^5 + 442*x^4 - 251*x^3 - 26*x^2 + 13*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 16 x^{18} + 7 x^{17} - 116 x^{16} + 257 x^{15} - 198 x^{14} - 275 x^{13} + 1033 x^{12} - 1484 x^{11} + 1114 x^{10} + 10 x^{9} - 849 x^{8} + 1469 x^{7} - 394 x^{6} + 261 x^{5} + 442 x^{4} - 251 x^{3} - 26 x^{2} + 13 x + 1 \)

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:  \(1537414689913008558084697161728=2^{16}\cdot 17\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.31$
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:  $2, 17, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{3}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{16} a^{12} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{8} a + \frac{7}{16}$, $\frac{1}{16} a^{13} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} - \frac{5}{16} a + \frac{1}{4}$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{1}{16} a^{11} + \frac{1}{16} a^{10} - \frac{1}{8} a^{8} + \frac{3}{16} a^{6} - \frac{3}{16} a^{5} + \frac{3}{16} a^{4} + \frac{11}{32} a^{2} + \frac{9}{32} a - \frac{5}{32}$, $\frac{1}{64} a^{15} - \frac{1}{32} a^{13} - \frac{1}{64} a^{12} - \frac{1}{16} a^{11} + \frac{3}{32} a^{10} - \frac{1}{8} a^{9} + \frac{3}{32} a^{7} - \frac{3}{16} a^{6} - \frac{3}{16} a^{5} + \frac{13}{32} a^{4} - \frac{9}{64} a^{3} - \frac{1}{4} a^{2} + \frac{3}{16} a + \frac{13}{64}$, $\frac{1}{256} a^{16} + \frac{1}{256} a^{15} + \frac{1}{128} a^{14} - \frac{7}{256} a^{13} + \frac{7}{256} a^{12} + \frac{5}{128} a^{11} + \frac{11}{128} a^{10} + \frac{3}{32} a^{9} + \frac{3}{128} a^{8} - \frac{11}{128} a^{7} - \frac{13}{128} a^{5} + \frac{121}{256} a^{4} + \frac{39}{256} a^{3} - \frac{5}{32} a^{2} + \frac{109}{256} a - \frac{55}{256}$, $\frac{1}{512} a^{17} + \frac{1}{512} a^{15} + \frac{7}{512} a^{14} - \frac{1}{256} a^{13} - \frac{13}{512} a^{12} - \frac{5}{128} a^{11} - \frac{15}{256} a^{10} - \frac{9}{256} a^{9} + \frac{9}{128} a^{8} + \frac{43}{256} a^{7} - \frac{29}{256} a^{6} - \frac{77}{512} a^{5} - \frac{25}{256} a^{4} - \frac{79}{512} a^{3} - \frac{187}{512} a^{2} - \frac{53}{128} a - \frac{153}{512}$, $\frac{1}{36864} a^{18} - \frac{1}{12288} a^{17} + \frac{53}{36864} a^{16} + \frac{1}{512} a^{15} + \frac{155}{12288} a^{14} + \frac{1037}{36864} a^{13} - \frac{913}{36864} a^{12} + \frac{403}{18432} a^{11} + \frac{103}{1152} a^{10} + \frac{253}{18432} a^{9} - \frac{623}{18432} a^{8} - \frac{1957}{9216} a^{7} - \frac{1055}{36864} a^{6} + \frac{5357}{36864} a^{5} + \frac{6203}{36864} a^{4} - \frac{1369}{18432} a^{3} + \frac{829}{36864} a^{2} - \frac{5299}{12288} a + \frac{13871}{36864}$, $\frac{1}{1253376} a^{19} - \frac{1}{104448} a^{18} - \frac{19}{19584} a^{17} + \frac{19}{139264} a^{16} - \frac{301}{417792} a^{15} + \frac{2417}{313344} a^{14} - \frac{16211}{626688} a^{13} + \frac{6863}{1253376} a^{12} - \frac{4571}{626688} a^{11} - \frac{57923}{626688} a^{10} + \frac{10255}{156672} a^{9} + \frac{65341}{626688} a^{8} + \frac{230965}{1253376} a^{7} - \frac{51223}{313344} a^{6} + \frac{67483}{626688} a^{5} + \frac{463003}{1253376} a^{4} - \frac{370385}{1253376} a^{3} + \frac{3299}{12288} a^{2} - \frac{38071}{78336} a + \frac{8225}{139264}$

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:  \( 175093980.343 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T513:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 128 conjugacy class representatives for t20n513 are not computed
Character table for t20n513 is not computed

Intermediate fields

\(\Q(\sqrt{53}) \), 5.5.2382032.1, 10.10.300726051798272.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$53$53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$