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Properties

Label 20.4.46080607852...3984.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{24}\cdot 13^{14}\cdot 17^{8}$
Root discriminant $42.97$
Ramified primes $2, 13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T226

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![118469, 139256, -60515, -122148, -13091, 54392, 25610, -13338, -19361, 5394, 5685, -844, -921, -778, 876, -236, 55, -66, 39, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 39*x^18 - 66*x^17 + 55*x^16 - 236*x^15 + 876*x^14 - 778*x^13 - 921*x^12 - 844*x^11 + 5685*x^10 + 5394*x^9 - 19361*x^8 - 13338*x^7 + 25610*x^6 + 54392*x^5 - 13091*x^4 - 122148*x^3 - 60515*x^2 + 139256*x + 118469)
 
gp: K = bnfinit(x^20 - 10*x^19 + 39*x^18 - 66*x^17 + 55*x^16 - 236*x^15 + 876*x^14 - 778*x^13 - 921*x^12 - 844*x^11 + 5685*x^10 + 5394*x^9 - 19361*x^8 - 13338*x^7 + 25610*x^6 + 54392*x^5 - 13091*x^4 - 122148*x^3 - 60515*x^2 + 139256*x + 118469, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 39 x^{18} - 66 x^{17} + 55 x^{16} - 236 x^{15} + 876 x^{14} - 778 x^{13} - 921 x^{12} - 844 x^{11} + 5685 x^{10} + 5394 x^{9} - 19361 x^{8} - 13338 x^{7} + 25610 x^{6} + 54392 x^{5} - 13091 x^{4} - 122148 x^{3} - 60515 x^{2} + 139256 x + 118469 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(460806078520572792990650790313984=2^{24}\cdot 13^{14}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{26} a^{14} + \frac{3}{13} a^{13} + \frac{5}{26} a^{12} - \frac{2}{13} a^{11} + \frac{3}{26} a^{10} - \frac{1}{2} a^{9} - \frac{6}{13} a^{8} + \frac{2}{13} a^{7} - \frac{3}{26} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{26} a^{15} - \frac{5}{26} a^{13} + \frac{5}{26} a^{12} + \frac{1}{26} a^{11} - \frac{5}{26} a^{10} + \frac{1}{26} a^{9} + \frac{11}{26} a^{8} - \frac{1}{26} a^{7} + \frac{5}{26} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{26} a^{16} - \frac{2}{13} a^{13} + \frac{1}{26} a^{11} + \frac{3}{26} a^{10} + \frac{11}{26} a^{9} - \frac{9}{26} a^{8} - \frac{1}{26} a^{7} + \frac{11}{26} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{26} a^{17} - \frac{1}{13} a^{13} - \frac{5}{26} a^{12} - \frac{3}{26} a^{10} - \frac{9}{26} a^{9} - \frac{5}{13} a^{8} - \frac{6}{13} a^{7} + \frac{1}{26} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{21260247573477770} a^{18} - \frac{9}{21260247573477770} a^{17} - \frac{307666483766233}{21260247573477770} a^{16} + \frac{8226380882633}{21260247573477770} a^{15} + \frac{11316292150447}{1932749779407070} a^{14} + \frac{19203325781137}{1932749779407070} a^{13} - \frac{1275526438927479}{10630123786738885} a^{12} - \frac{4646047329017007}{21260247573477770} a^{11} - \frac{1691187475481379}{21260247573477770} a^{10} - \frac{7727891171896157}{21260247573477770} a^{9} + \frac{132884949198388}{2126024757347777} a^{8} + \frac{1814140120482127}{4252049514695554} a^{7} + \frac{417449176189172}{10630123786738885} a^{6} - \frac{30271091720169}{74336529977195} a^{5} + \frac{330440908386873}{1635403659498290} a^{4} - \frac{394368099165732}{817701829749145} a^{3} + \frac{69942265820969}{163540365949829} a^{2} + \frac{9161821957538}{817701829749145} a + \frac{402387869529673}{817701829749145}$, $\frac{1}{8675945610527528814910} a^{19} + \frac{102016}{4337972805263764407455} a^{18} - \frac{51612829458421885857}{8675945610527528814910} a^{17} - \frac{12460300413509791837}{867594561052752881491} a^{16} - \frac{1506952541603501302}{867594561052752881491} a^{15} - \frac{2786242004231739421}{788722328229775346810} a^{14} - \frac{305132098205865109753}{4337972805263764407455} a^{13} - \frac{16322941767212663430}{66738043157904067807} a^{12} - \frac{24229926601305831548}{4337972805263764407455} a^{11} + \frac{1030124266379776369999}{8675945610527528814910} a^{10} - \frac{2277547979988177580627}{8675945610527528814910} a^{9} + \frac{214897084694569604385}{867594561052752881491} a^{8} - \frac{62608320816958461197}{667380431579040678070} a^{7} - \frac{320958653692298029}{10267391255062164278} a^{6} + \frac{20168248873858960877}{66738043157904067807} a^{5} + \frac{109466929492178475117}{333690215789520339035} a^{4} - \frac{122502509572460315182}{333690215789520339035} a^{3} - \frac{63033837255568344567}{333690215789520339035} a^{2} + \frac{23501145598325099319}{51336956275310821390} a + \frac{4200960597733673361}{25668478137655410695}$

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:  $11$
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:  \( 422552818.542 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T226:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1920
The 18 conjugacy class representatives for t20n226
Character table for t20n226

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.10158928.1, 10.10.1341649635419392.1, 10.2.1651261089746944.1, 10.2.21466394166710272.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$