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Properties

Label 20.0.55338861907...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 11^{2}\cdot 28162171^{2}$
Root discriminant $27.36$
Ramified primes $3, 5, 11, 28162171$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group 20T656

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -13, 149, -316, 810, -659, 1473, -802, 1882, -554, 1455, -320, 804, -114, 289, -35, 74, -6, 11, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 11*x^18 - 6*x^17 + 74*x^16 - 35*x^15 + 289*x^14 - 114*x^13 + 804*x^12 - 320*x^11 + 1455*x^10 - 554*x^9 + 1882*x^8 - 802*x^7 + 1473*x^6 - 659*x^5 + 810*x^4 - 316*x^3 + 149*x^2 - 13*x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 11*x^18 - 6*x^17 + 74*x^16 - 35*x^15 + 289*x^14 - 114*x^13 + 804*x^12 - 320*x^11 + 1455*x^10 - 554*x^9 + 1882*x^8 - 802*x^7 + 1473*x^6 - 659*x^5 + 810*x^4 - 316*x^3 + 149*x^2 - 13*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 11 x^{18} - 6 x^{17} + 74 x^{16} - 35 x^{15} + 289 x^{14} - 114 x^{13} + 804 x^{12} - 320 x^{11} + 1455 x^{10} - 554 x^{9} + 1882 x^{8} - 802 x^{7} + 1473 x^{6} - 659 x^{5} + 810 x^{4} - 316 x^{3} + 149 x^{2} - 13 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(55338861907337413914931640625=3^{10}\cdot 5^{10}\cdot 11^{2}\cdot 28162171^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.36$
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:  $3, 5, 11, 28162171$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{5914063439482232170273} a^{19} + \frac{508564259256270928225}{5914063439482232170273} a^{18} + \frac{762928901780689469178}{5914063439482232170273} a^{17} - \frac{2725000235762912546841}{5914063439482232170273} a^{16} + \frac{1502734690137193324956}{5914063439482232170273} a^{15} - \frac{39204905829430612809}{100238363381054782547} a^{14} + \frac{525231261511507633520}{5914063439482232170273} a^{13} - \frac{2534537219843578346069}{5914063439482232170273} a^{12} + \frac{40361391557873575518}{100238363381054782547} a^{11} + \frac{2581090287555851262653}{5914063439482232170273} a^{10} + \frac{2408891461460866383676}{5914063439482232170273} a^{9} - \frac{1346226275787664920566}{5914063439482232170273} a^{8} + \frac{2556070711619077055636}{5914063439482232170273} a^{7} + \frac{2077828808367992184730}{5914063439482232170273} a^{6} - \frac{597442317661444191345}{5914063439482232170273} a^{5} + \frac{1240586495591282745183}{5914063439482232170273} a^{4} + \frac{729872926286632343414}{5914063439482232170273} a^{3} - \frac{1562973207572770940005}{5914063439482232170273} a^{2} - \frac{489235854346318215229}{5914063439482232170273} a + \frac{1661834536985280973391}{5914063439482232170273}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{481423089498775426300}{5914063439482232170273} a^{19} + \frac{457190000518990909386}{5914063439482232170273} a^{18} - \frac{5297374111420868654026}{5914063439482232170273} a^{17} + \frac{2649429174284105792770}{5914063439482232170273} a^{16} - \frac{35758443794612160800858}{5914063439482232170273} a^{15} + \frac{257835527382490076617}{100238363381054782547} a^{14} - \frac{140118793239800620111586}{5914063439482232170273} a^{13} + \frac{48774170586092721151689}{5914063439482232170273} a^{12} - \frac{6631523845893852775211}{100238363381054782547} a^{11} + \frac{137182616469751221277855}{5914063439482232170273} a^{10} - \frac{711522917802905591599240}{5914063439482232170273} a^{9} + \frac{238354831737938217275281}{5914063439482232170273} a^{8} - \frac{924986638690453819156195}{5914063439482232170273} a^{7} + \frac{349995636052071441388157}{5914063439482232170273} a^{6} - \frac{729327018074998426895926}{5914063439482232170273} a^{5} + \frac{294389079334357416430156}{5914063439482232170273} a^{4} - \frac{401340250835521085453693}{5914063439482232170273} a^{3} + \frac{136117002773111363379129}{5914063439482232170273} a^{2} - \frac{77174788362646691854509}{5914063439482232170273} a + \frac{6743917501260964804378}{5914063439482232170273} \) (order $6$)
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:  \( 170321.970246 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T656:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 70 conjugacy class representatives for t20n656 are not computed
Character table for t20n656 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 10.10.968074628125.1

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 24 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5Data not computed
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
28162171Data not computed