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Properties

Label 20.8.17533216858...3125.2
Degree $20$
Signature $[8, 6]$
Discriminant $3^{10}\cdot 5^{11}\cdot 239^{10}$
Root discriminant $64.89$
Ramified primes $3, 5, 239$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-417919, 6355544, -13841592, -3483430, 31934522, -33069267, 16577353, -7442088, 3406720, -1064806, 454875, -229690, 80512, -33416, 12198, -3205, 945, -226, 38, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 38*x^18 - 226*x^17 + 945*x^16 - 3205*x^15 + 12198*x^14 - 33416*x^13 + 80512*x^12 - 229690*x^11 + 454875*x^10 - 1064806*x^9 + 3406720*x^8 - 7442088*x^7 + 16577353*x^6 - 33069267*x^5 + 31934522*x^4 - 3483430*x^3 - 13841592*x^2 + 6355544*x - 417919)
 
gp: K = bnfinit(x^20 - 8*x^19 + 38*x^18 - 226*x^17 + 945*x^16 - 3205*x^15 + 12198*x^14 - 33416*x^13 + 80512*x^12 - 229690*x^11 + 454875*x^10 - 1064806*x^9 + 3406720*x^8 - 7442088*x^7 + 16577353*x^6 - 33069267*x^5 + 31934522*x^4 - 3483430*x^3 - 13841592*x^2 + 6355544*x - 417919, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 38 x^{18} - 226 x^{17} + 945 x^{16} - 3205 x^{15} + 12198 x^{14} - 33416 x^{13} + 80512 x^{12} - 229690 x^{11} + 454875 x^{10} - 1064806 x^{9} + 3406720 x^{8} - 7442088 x^{7} + 16577353 x^{6} - 33069267 x^{5} + 31934522 x^{4} - 3483430 x^{3} - 13841592 x^{2} + 6355544 x - 417919 \)

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:  \(1753321685810638349237472141064453125=3^{10}\cdot 5^{11}\cdot 239^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.89$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{2}{9} a^{11} - \frac{2}{9} a^{10} + \frac{4}{9} a^{9} + \frac{1}{3} a^{8} - \frac{2}{9} a^{7} - \frac{2}{9} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{2}{9} a^{2} - \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{14} - \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{2}{9} a^{11} - \frac{2}{9} a^{10} + \frac{1}{3} a^{9} + \frac{1}{9} a^{8} + \frac{4}{9} a^{7} - \frac{1}{3} a^{4} - \frac{1}{9} a^{3} + \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{14} - \frac{1}{9} a^{12} - \frac{4}{9} a^{10} - \frac{1}{3} a^{9} + \frac{1}{9} a^{8} + \frac{2}{9} a^{7} + \frac{2}{9} a^{6} + \frac{2}{9} a^{4} + \frac{1}{3} a^{3} - \frac{1}{9} a^{2} - \frac{1}{9}$, $\frac{1}{44631} a^{18} - \frac{112}{14877} a^{17} + \frac{1696}{44631} a^{16} - \frac{1751}{44631} a^{15} + \frac{7063}{44631} a^{14} - \frac{2159}{14877} a^{13} + \frac{6788}{44631} a^{12} + \frac{17549}{44631} a^{11} - \frac{761}{4959} a^{10} + \frac{728}{1653} a^{9} - \frac{3676}{14877} a^{8} + \frac{13910}{44631} a^{7} + \frac{21200}{44631} a^{6} - \frac{308}{1539} a^{5} + \frac{5614}{14877} a^{4} + \frac{17128}{44631} a^{3} + \frac{9760}{44631} a^{2} + \frac{2899}{14877} a + \frac{529}{1539}$, $\frac{1}{77540301613277196476747162866705160670162956180112054584395029} a^{19} + \frac{725009322489477956910996307162541206109428267092819078940}{77540301613277196476747162866705160670162956180112054584395029} a^{18} + \frac{4478187084028909202279456659285921881658103672102839649195}{120967709225081429760916010712488550187461710109379180318869} a^{17} + \frac{2293212565310576372019626971987772389059390526716550874509394}{77540301613277196476747162866705160670162956180112054584395029} a^{16} - \frac{3520998513563703303867182014372539206388741224808387825560869}{77540301613277196476747162866705160670162956180112054584395029} a^{15} + \frac{8966603629188015177586665749161752449168974386892686227797}{116252326256787401014613437581267107451518674932701731011087} a^{14} - \frac{9700654969466488057436291837010317376999687441944762329489084}{77540301613277196476747162866705160670162956180112054584395029} a^{13} - \frac{4221341651328282186795589765702306681176755047573929258348545}{77540301613277196476747162866705160670162956180112054584395029} a^{12} + \frac{35932740512858637476405519776866288391280396124375138242337800}{77540301613277196476747162866705160670162956180112054584395029} a^{11} + \frac{640239427088833358627544453920937259724857334099672835666464}{8615589068141910719638573651856128963351439575568006064932781} a^{10} - \frac{9853928174309750144632291069482666687904307563718040050219191}{25846767204425732158915720955568386890054318726704018194798343} a^{9} - \frac{10826403116352387014748254567015719954037148319092461122060756}{77540301613277196476747162866705160670162956180112054584395029} a^{8} - \frac{13417693377849345561447214343082463272781710371760446455592678}{77540301613277196476747162866705160670162956180112054584395029} a^{7} + \frac{26338400708799621475507656512742071695332950901127824323361166}{77540301613277196476747162866705160670162956180112054584395029} a^{6} + \frac{9188062839520675298967862398744904281819402178520469611288030}{77540301613277196476747162866705160670162956180112054584395029} a^{5} + \frac{22905722102150522868174967777647677085495387249648267484683910}{77540301613277196476747162866705160670162956180112054584395029} a^{4} - \frac{10393813080996267579260450378708786983481822258438479383362359}{77540301613277196476747162866705160670162956180112054584395029} a^{3} + \frac{27839894869642235505440149518420692717739047460502709925197836}{77540301613277196476747162866705160670162956180112054584395029} a^{2} - \frac{4784409678545119708820270869659109034182787931408919047906631}{77540301613277196476747162866705160670162956180112054584395029} a + \frac{42210583657029332589468324484188809769064909612064739572056}{140726500205584748596637319177323340599206817023796832276579}$

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:  $13$
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:  \( 33308369752.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 $20$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed