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Properties

Label 20.16.8274349147...4752.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{20}\cdot 13^{16}\cdot 17^{9}$
Root discriminant $55.70$
Ramified primes $2, 13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T803

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, -1274, 127, 9548, -3927, -23726, 15716, 19318, -22931, 4578, 12289, -12082, -2155, 5810, -352, -1270, 205, 124, -27, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 27*x^18 + 124*x^17 + 205*x^16 - 1270*x^15 - 352*x^14 + 5810*x^13 - 2155*x^12 - 12082*x^11 + 12289*x^10 + 4578*x^9 - 22931*x^8 + 19318*x^7 + 15716*x^6 - 23726*x^5 - 3927*x^4 + 9548*x^3 + 127*x^2 - 1274*x + 49)
 
gp: K = bnfinit(x^20 - 4*x^19 - 27*x^18 + 124*x^17 + 205*x^16 - 1270*x^15 - 352*x^14 + 5810*x^13 - 2155*x^12 - 12082*x^11 + 12289*x^10 + 4578*x^9 - 22931*x^8 + 19318*x^7 + 15716*x^6 - 23726*x^5 - 3927*x^4 + 9548*x^3 + 127*x^2 - 1274*x + 49, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 27 x^{18} + 124 x^{17} + 205 x^{16} - 1270 x^{15} - 352 x^{14} + 5810 x^{13} - 2155 x^{12} - 12082 x^{11} + 12289 x^{10} + 4578 x^{9} - 22931 x^{8} + 19318 x^{7} + 15716 x^{6} - 23726 x^{5} - 3927 x^{4} + 9548 x^{3} + 127 x^{2} - 1274 x + 49 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(82743491474350352141383732535754752=2^{20}\cdot 13^{16}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.70$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7} a^{17} + \frac{1}{7} a^{16} - \frac{3}{7} a^{15} + \frac{2}{7} a^{14} - \frac{3}{7} a^{13} - \frac{1}{7} a^{12} - \frac{1}{7} a^{11} - \frac{3}{7} a^{10} + \frac{2}{7} a^{9} + \frac{2}{7} a^{8} + \frac{3}{7} a^{7} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{7} a^{18} + \frac{3}{7} a^{16} - \frac{2}{7} a^{15} + \frac{2}{7} a^{14} + \frac{2}{7} a^{13} - \frac{2}{7} a^{11} - \frac{2}{7} a^{10} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7} a$, $\frac{1}{65202455381494346364471360471175} a^{19} + \frac{702458565340925108287500906839}{65202455381494346364471360471175} a^{18} + \frac{45054841817451250778218054804}{2608098215259773854578854418847} a^{17} + \frac{6231881252984888222009802802924}{65202455381494346364471360471175} a^{16} + \frac{24566652440589625066631206191887}{65202455381494346364471360471175} a^{15} + \frac{4230009868459857370819945656253}{9314636483070620909210194353025} a^{14} + \frac{21367874273502040899797158602326}{65202455381494346364471360471175} a^{13} - \frac{1095687921460971671482526479371}{9314636483070620909210194353025} a^{12} + \frac{23479543157694799278994372817249}{65202455381494346364471360471175} a^{11} + \frac{341428113392822280009955377714}{2608098215259773854578854418847} a^{10} + \frac{22320948561018786006192558788539}{65202455381494346364471360471175} a^{9} - \frac{6254455877144120592937714358074}{13040491076298869272894272094235} a^{8} - \frac{4962192387329861350188998919}{167615566533404489368820978075} a^{7} - \frac{4888465390937812845794797046689}{13040491076298869272894272094235} a^{6} + \frac{18640261587601406911580712327981}{65202455381494346364471360471175} a^{5} + \frac{31812820637041410982583843720757}{65202455381494346364471360471175} a^{4} + \frac{5983702265347435576028891972874}{65202455381494346364471360471175} a^{3} + \frac{1763980882605163395697223625101}{13040491076298869272894272094235} a^{2} - \frac{29256153032378620341909357475458}{65202455381494346364471360471175} a - \frac{1196278593260273646874442214649}{9314636483070620909210194353025}$

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:  $17$
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:  \( 69064961785.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T803:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 126 conjugacy class representatives for t20n803 are not computed
Character table for t20n803 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.10158928.1, 10.10.1341649635419392.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.12.11.6$x^{12} - 13312$$12$$1$$11$$C_{12}$$[\ ]_{12}$
$17$17.2.0.1$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$$[\ ]^{4}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.5.1$x^{6} - 17$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$