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Properties

Label 20.8.39780524747...5744.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{16}\cdot 13^{15}\cdot 17^{9}$
Root discriminant $42.66$
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![-6877, 165945, 41795, -245479, 1378, 96317, -79677, 64675, -9126, -3575, 1105, -6435, 1264, 1945, 117, -519, -52, 89, -1, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 - x^18 + 89*x^17 - 52*x^16 - 519*x^15 + 117*x^14 + 1945*x^13 + 1264*x^12 - 6435*x^11 + 1105*x^10 - 3575*x^9 - 9126*x^8 + 64675*x^7 - 79677*x^6 + 96317*x^5 + 1378*x^4 - 245479*x^3 + 41795*x^2 + 165945*x - 6877)
 
gp: K = bnfinit(x^20 - 7*x^19 - x^18 + 89*x^17 - 52*x^16 - 519*x^15 + 117*x^14 + 1945*x^13 + 1264*x^12 - 6435*x^11 + 1105*x^10 - 3575*x^9 - 9126*x^8 + 64675*x^7 - 79677*x^6 + 96317*x^5 + 1378*x^4 - 245479*x^3 + 41795*x^2 + 165945*x - 6877, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} - x^{18} + 89 x^{17} - 52 x^{16} - 519 x^{15} + 117 x^{14} + 1945 x^{13} + 1264 x^{12} - 6435 x^{11} + 1105 x^{10} - 3575 x^{9} - 9126 x^{8} + 64675 x^{7} - 79677 x^{6} + 96317 x^{5} + 1378 x^{4} - 245479 x^{3} + 41795 x^{2} + 165945 x - 6877 \)

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:  \(397805247472838231448960252575744=2^{16}\cdot 13^{15}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.66$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{17} a^{17} + \frac{8}{17} a^{16} - \frac{7}{17} a^{15} + \frac{2}{17} a^{14} + \frac{7}{17} a^{13} + \frac{6}{17} a^{12} + \frac{7}{17} a^{10} - \frac{6}{17} a^{9} + \frac{5}{17} a^{8} + \frac{6}{17} a^{7} - \frac{3}{17} a^{6} - \frac{3}{17} a^{5} + \frac{2}{17} a^{4} + \frac{1}{17} a^{3} - \frac{5}{17} a^{2} - \frac{1}{17} a - \frac{7}{17}$, $\frac{1}{221} a^{18} + \frac{3}{221} a^{17} + \frac{4}{221} a^{16} + \frac{54}{221} a^{15} - \frac{54}{221} a^{14} + \frac{22}{221} a^{13} - \frac{81}{221} a^{12} - \frac{6}{17} a^{11} + \frac{6}{17} a^{10} + \frac{4}{17} a^{9} - \frac{8}{17} a^{8} + \frac{4}{17} a^{7} - \frac{3}{17} a^{6} - \frac{2}{17} a^{4} - \frac{6}{17} a^{3} - \frac{6}{17} a^{2} - \frac{8}{17} a + \frac{4}{17}$, $\frac{1}{62844185480094876391603512794706776535770747} a^{19} + \frac{26088420655862575457836239884783207851807}{62844185480094876391603512794706776535770747} a^{18} + \frac{105469471259406888457846945764823098585979}{8977740782870696627371930399243825219395821} a^{17} - \frac{450040600012683797656812522642709894055502}{3696716792946757434800206634982751560927691} a^{16} + \frac{11587935597653156007916107811394325710041331}{62844185480094876391603512794706776535770747} a^{15} + \frac{21138281348483854915243362017929479589733031}{62844185480094876391603512794706776535770747} a^{14} - \frac{9227385495222870419056707009900993321730257}{62844185480094876391603512794706776535770747} a^{13} + \frac{2214271748452984167014656082624063595842962}{62844185480094876391603512794706776535770747} a^{12} + \frac{584999189522723508121573746503945151072304}{4834168113853452030123347138054367425828519} a^{11} - \frac{84246383086438154255405118148673292303204}{690595444836207432874763876864909632261217} a^{10} + \frac{1321111204390384539781171436240673919077548}{4834168113853452030123347138054367425828519} a^{9} - \frac{646717133705789721653849465927800890428478}{4834168113853452030123347138054367425828519} a^{8} - \frac{1842556541823866124745436557223353676536029}{4834168113853452030123347138054367425828519} a^{7} + \frac{2170599473317667571795848237073645794994352}{4834168113853452030123347138054367425828519} a^{6} + \frac{868746034237657033515755720552449250527782}{4834168113853452030123347138054367425828519} a^{5} - \frac{1187613686455690187018782949078208036140410}{4834168113853452030123347138054367425828519} a^{4} - \frac{1380128079980855020163719805683415443670384}{4834168113853452030123347138054367425828519} a^{3} + \frac{88791539060209125792074144832631892084580}{210181222341454436092319440784972496775153} a^{2} + \frac{179534075090790458468623446958064046127699}{690595444836207432874763876864909632261217} a - \frac{2896956371864235283679713651641112736885}{210181222341454436092319440784972496775153}$

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:  \( 790471653.551 \) (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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
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}$
13.12.11.3$x^{12} - 208$$12$$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$$[\ ]^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.6.5.2$x^{6} + 51$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$