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Properties

Label 20.12.6634545047...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{10}\cdot 79^{5}\cdot 4588681^{2}$
Root discriminant $61.81$
Ramified primes $2, 5, 79, 4588681$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1903025191, -508131864, -2885229281, 526391474, 1754967268, -227369546, -571168620, 62059998, 117660653, -10746878, -16443936, 1155418, 1581705, -74430, -102439, 2612, 4222, -38, -99, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 99*x^18 - 38*x^17 + 4222*x^16 + 2612*x^15 - 102439*x^14 - 74430*x^13 + 1581705*x^12 + 1155418*x^11 - 16443936*x^10 - 10746878*x^9 + 117660653*x^8 + 62059998*x^7 - 571168620*x^6 - 227369546*x^5 + 1754967268*x^4 + 526391474*x^3 - 2885229281*x^2 - 508131864*x + 1903025191)
 
gp: K = bnfinit(x^20 - 99*x^18 - 38*x^17 + 4222*x^16 + 2612*x^15 - 102439*x^14 - 74430*x^13 + 1581705*x^12 + 1155418*x^11 - 16443936*x^10 - 10746878*x^9 + 117660653*x^8 + 62059998*x^7 - 571168620*x^6 - 227369546*x^5 + 1754967268*x^4 + 526391474*x^3 - 2885229281*x^2 - 508131864*x + 1903025191, 1)
 

Normalized defining polynomial

\( x^{20} - 99 x^{18} - 38 x^{17} + 4222 x^{16} + 2612 x^{15} - 102439 x^{14} - 74430 x^{13} + 1581705 x^{12} + 1155418 x^{11} - 16443936 x^{10} - 10746878 x^{9} + 117660653 x^{8} + 62059998 x^{7} - 571168620 x^{6} - 227369546 x^{5} + 1754967268 x^{4} + 526391474 x^{3} - 2885229281 x^{2} - 508131864 x + 1903025191 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(663454504774367623174543360000000000=2^{20}\cdot 5^{10}\cdot 79^{5}\cdot 4588681^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.81$
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:  $2, 5, 79, 4588681$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{59} a^{18} + \frac{10}{59} a^{17} + \frac{3}{59} a^{16} + \frac{12}{59} a^{15} - \frac{18}{59} a^{14} - \frac{22}{59} a^{13} + \frac{24}{59} a^{12} - \frac{12}{59} a^{11} + \frac{20}{59} a^{10} + \frac{20}{59} a^{9} + \frac{17}{59} a^{8} + \frac{5}{59} a^{7} - \frac{13}{59} a^{6} + \frac{20}{59} a^{5} - \frac{7}{59} a^{4} + \frac{22}{59} a^{3} + \frac{25}{59} a^{2} + \frac{22}{59} a + \frac{22}{59}$, $\frac{1}{80219986723186809610075105108900438769807378350618594527825943361136591419} a^{19} - \frac{377666885982095950098094299956003190846631924756721639543948308376656171}{80219986723186809610075105108900438769807378350618594527825943361136591419} a^{18} + \frac{21489309016815156615158626957062806907221072735637454939903640594237137372}{80219986723186809610075105108900438769807378350618594527825943361136591419} a^{17} - \frac{22388033335335957358777814303314892081403352592724343701386957014783846130}{80219986723186809610075105108900438769807378350618594527825943361136591419} a^{16} + \frac{19734163473254822993367330764141575270800521755585217194747177148870228289}{80219986723186809610075105108900438769807378350618594527825943361136591419} a^{15} + \frac{13479509913868457857716463247219318445101374259286024849354163090528076614}{80219986723186809610075105108900438769807378350618594527825943361136591419} a^{14} + \frac{10717009608777808485428636469283735524128786597054434862616936261559507389}{80219986723186809610075105108900438769807378350618594527825943361136591419} a^{13} - \frac{7016657836023180095686386469370577605724011735610411737435738547251765396}{80219986723186809610075105108900438769807378350618594527825943361136591419} a^{12} - \frac{18542809798340591146633757429182814317706028814589579914497268993848148821}{80219986723186809610075105108900438769807378350618594527825943361136591419} a^{11} - \frac{36097097421702547930219176951736129109197355211149484569398719646416227218}{80219986723186809610075105108900438769807378350618594527825943361136591419} a^{10} + \frac{16891563862258301423726332900499521734352991456816792381585790265865045732}{80219986723186809610075105108900438769807378350618594527825943361136591419} a^{9} + \frac{29543718706554905505848435161623554857157094249831883424757422240895758368}{80219986723186809610075105108900438769807378350618594527825943361136591419} a^{8} - \frac{33526057050361863960664372310056668445592018477944302140891997107755508728}{80219986723186809610075105108900438769807378350618594527825943361136591419} a^{7} - \frac{24282983575640794545137680678415013718333189238663276406263930363563887740}{80219986723186809610075105108900438769807378350618594527825943361136591419} a^{6} - \frac{16879858362697040338965378087957472286920056590361309248495380545808925895}{80219986723186809610075105108900438769807378350618594527825943361136591419} a^{5} - \frac{34709501796558930905808522779947455077419679823728843479568742400026271849}{80219986723186809610075105108900438769807378350618594527825943361136591419} a^{4} - \frac{24456367861062093498854318608958123890476964148926252758878890735125293383}{80219986723186809610075105108900438769807378350618594527825943361136591419} a^{3} - \frac{11701865721359650548021971393823669920072010328432808355520555813447046282}{80219986723186809610075105108900438769807378350618594527825943361136591419} a^{2} + \frac{9552404833137797471568536149417588000600491584112644517643631434844778572}{80219986723186809610075105108900438769807378350618594527825943361136591419} a - \frac{1064468454212273106787760783268231574806657365729760527964909767091499245}{80219986723186809610075105108900438769807378350618594527825943361136591419}$

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:  $15$
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:  \( 42807576702.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 115200
The 119 conjugacy class representatives for t20n781 are not computed
Character table for t20n781 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.31600.1, 10.6.14339628125.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
5Data not computed
79Data not computed
4588681Data not computed