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Properties

Label 20.4.79144095554...3472.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{34}\cdot 7^{11}\cdot 13^{12}$
Root discriminant $44.15$
Ramified primes $2, 7, 13$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T633

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1559, -2748, -3848, -392, 5906, 11246, 12800, 10942, 14353, 9562, -1617, -3720, -2221, -1110, -43, 122, 66, -14, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 12*x^18 - 14*x^17 + 66*x^16 + 122*x^15 - 43*x^14 - 1110*x^13 - 2221*x^12 - 3720*x^11 - 1617*x^10 + 9562*x^9 + 14353*x^8 + 10942*x^7 + 12800*x^6 + 11246*x^5 + 5906*x^4 - 392*x^3 - 3848*x^2 - 2748*x - 1559)
 
gp: K = bnfinit(x^20 - 12*x^18 - 14*x^17 + 66*x^16 + 122*x^15 - 43*x^14 - 1110*x^13 - 2221*x^12 - 3720*x^11 - 1617*x^10 + 9562*x^9 + 14353*x^8 + 10942*x^7 + 12800*x^6 + 11246*x^5 + 5906*x^4 - 392*x^3 - 3848*x^2 - 2748*x - 1559, 1)
 

Normalized defining polynomial

\( x^{20} - 12 x^{18} - 14 x^{17} + 66 x^{16} + 122 x^{15} - 43 x^{14} - 1110 x^{13} - 2221 x^{12} - 3720 x^{11} - 1617 x^{10} + 9562 x^{9} + 14353 x^{8} + 10942 x^{7} + 12800 x^{6} + 11246 x^{5} + 5906 x^{4} - 392 x^{3} - 3848 x^{2} - 2748 x - 1559 \)

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:  \(791440955544624095439021483753472=2^{34}\cdot 7^{11}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.15$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} - \frac{2}{7} a^{13} + \frac{2}{7} a^{12} + \frac{3}{7} a^{11} + \frac{3}{7} a^{10} - \frac{3}{7} a^{9} + \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{15} - \frac{2}{7} a^{13} + \frac{2}{7} a^{11} + \frac{3}{7} a^{10} + \frac{1}{7} a^{9} + \frac{2}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{16} + \frac{3}{7} a^{13} - \frac{1}{7} a^{12} + \frac{2}{7} a^{11} + \frac{1}{7} a^{9} + \frac{2}{7} a^{8} + \frac{2}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{17} - \frac{2}{7} a^{13} + \frac{3}{7} a^{12} - \frac{2}{7} a^{11} - \frac{1}{7} a^{10} - \frac{3}{7} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{3}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{11417} a^{18} - \frac{87}{11417} a^{17} + \frac{52}{1631} a^{16} + \frac{425}{11417} a^{15} + \frac{107}{11417} a^{14} + \frac{68}{11417} a^{13} + \frac{746}{11417} a^{12} - \frac{596}{11417} a^{11} - \frac{2549}{11417} a^{10} - \frac{1382}{11417} a^{9} + \frac{440}{11417} a^{8} + \frac{3683}{11417} a^{7} + \frac{1411}{11417} a^{6} + \frac{1238}{11417} a^{5} - \frac{3180}{11417} a^{4} - \frac{531}{1631} a^{3} + \frac{3681}{11417} a^{2} + \frac{4959}{11417} a + \frac{5310}{11417}$, $\frac{1}{1871592219684864476564269008433074445013} a^{19} - \frac{43207788344533582468297747382966583}{1871592219684864476564269008433074445013} a^{18} - \frac{52061169068750769326178210838999802680}{1871592219684864476564269008433074445013} a^{17} - \frac{18415927881860121870690717676150954162}{1871592219684864476564269008433074445013} a^{16} - \frac{90222569790898017064258328922491426329}{1871592219684864476564269008433074445013} a^{15} - \frac{46266722086206893024965121502880144940}{1871592219684864476564269008433074445013} a^{14} - \frac{687056471241743534736867671133023446709}{1871592219684864476564269008433074445013} a^{13} - \frac{451747535094240134546596820874398049710}{1871592219684864476564269008433074445013} a^{12} - \frac{3286562197890926147142672437998174078}{12394650461489168718968668930020360563} a^{11} + \frac{508931194364728162841940009914540964040}{1871592219684864476564269008433074445013} a^{10} - \frac{451321149244933037262669619531728218035}{1871592219684864476564269008433074445013} a^{9} - \frac{914237600727150723291701846044682399360}{1871592219684864476564269008433074445013} a^{8} - \frac{742312531611882651535919420738096638381}{1871592219684864476564269008433074445013} a^{7} + \frac{15070564569082664566593139313182395103}{50583573504996337204439702930623633649} a^{6} + \frac{61297822911439920663665600897705965200}{267370317097837782366324144061867777859} a^{5} - \frac{923482693113041312010288849584626328522}{1871592219684864476564269008433074445013} a^{4} - \frac{448653949353908920198347952664730434174}{1871592219684864476564269008433074445013} a^{3} - \frac{663153522432581244968706701505472619546}{1871592219684864476564269008433074445013} a^{2} + \frac{45675347356700313009133969193072973037}{1871592219684864476564269008433074445013} a + \frac{6660195247157920091882967267983335589}{1871592219684864476564269008433074445013}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 285555310.173 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T633:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40960
The 124 conjugacy class representatives for t20n633 are not computed
Character table for t20n633 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.5.6889792.1, 10.10.379753870426112.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.6.3$x^{8} - 7 x^{4} + 147$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$