↑

Properties

Label 20.0.27043715002...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{10}\cdot 5^{12}\cdot 17^{15}$
Root discriminant $66.31$
Ramified primes $2, 3, 5, 17$
Class number $20$ (GRH)
Class group $[2, 10]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![435562168, -233337512, 565677856, -276077824, 333088280, -139838948, 113547748, -39678884, 24673492, -6995232, 3601118, -806086, 366942, -62922, 27090, -3449, 1471, -134, 54, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 54*x^18 - 134*x^17 + 1471*x^16 - 3449*x^15 + 27090*x^14 - 62922*x^13 + 366942*x^12 - 806086*x^11 + 3601118*x^10 - 6995232*x^9 + 24673492*x^8 - 39678884*x^7 + 113547748*x^6 - 139838948*x^5 + 333088280*x^4 - 276077824*x^3 + 565677856*x^2 - 233337512*x + 435562168)
 
gp: K = bnfinit(x^20 - 3*x^19 + 54*x^18 - 134*x^17 + 1471*x^16 - 3449*x^15 + 27090*x^14 - 62922*x^13 + 366942*x^12 - 806086*x^11 + 3601118*x^10 - 6995232*x^9 + 24673492*x^8 - 39678884*x^7 + 113547748*x^6 - 139838948*x^5 + 333088280*x^4 - 276077824*x^3 + 565677856*x^2 - 233337512*x + 435562168, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 54 x^{18} - 134 x^{17} + 1471 x^{16} - 3449 x^{15} + 27090 x^{14} - 62922 x^{13} + 366942 x^{12} - 806086 x^{11} + 3601118 x^{10} - 6995232 x^{9} + 24673492 x^{8} - 39678884 x^{7} + 113547748 x^{6} - 139838948 x^{5} + 333088280 x^{4} - 276077824 x^{3} + 565677856 x^{2} - 233337512 x + 435562168 \)

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:  \(2704371500297649804173712000000000000=2^{16}\cdot 3^{10}\cdot 5^{12}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.31$
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:  $2, 3, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{14} - \frac{1}{4} a^{13} + \frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{14} + \frac{1}{8} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{8} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{18} - \frac{1}{4} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{3923998895179190410054154942878373334753080506548210522040974136} a^{19} + \frac{16777677703821201523839583132683960556472778445999363414385675}{980999723794797602513538735719593333688270126637052630510243534} a^{18} + \frac{11869547547860001001571791049043464984346850899173003553092435}{490499861897398801256769367859796666844135063318526315255121767} a^{17} + \frac{80713968518038119537964407374437434034977890199113524588331941}{1961999447589595205027077471439186667376540253274105261020487068} a^{16} + \frac{152459751389942902150543492566117991625632159576400592979955073}{3923998895179190410054154942878373334753080506548210522040974136} a^{15} - \frac{37790541461997067607737710212860680281918174785632601707024209}{3923998895179190410054154942878373334753080506548210522040974136} a^{14} - \frac{25901206501167253498063826717448818422562372748570461787134219}{980999723794797602513538735719593333688270126637052630510243534} a^{13} + \frac{400971907189360424395618474189330963130997450185928846076785291}{1961999447589595205027077471439186667376540253274105261020487068} a^{12} - \frac{411055147572254369886998275267622250946367510361589403032439105}{1961999447589595205027077471439186667376540253274105261020487068} a^{11} - \frac{930315656970260193634977248456230098588453237571487092641349877}{3923998895179190410054154942878373334753080506548210522040974136} a^{10} - \frac{382515415988900289875053683144302030380997634518506009611666325}{980999723794797602513538735719593333688270126637052630510243534} a^{9} - \frac{480083029744293106675781891152832931464121969893201679562593697}{1961999447589595205027077471439186667376540253274105261020487068} a^{8} - \frac{382834787425254064233239648784489342860337927281963990479772609}{980999723794797602513538735719593333688270126637052630510243534} a^{7} - \frac{565311837171835542165438900623589260629036294099158306662999807}{1961999447589595205027077471439186667376540253274105261020487068} a^{6} + \frac{70555261832011972295496302979453515787623380627527877085747431}{490499861897398801256769367859796666844135063318526315255121767} a^{5} + \frac{31499270117489020172302172149428270011888885358987900134523801}{490499861897398801256769367859796666844135063318526315255121767} a^{4} - \frac{208977840637117981324882258366646718472172425226341006860027522}{490499861897398801256769367859796666844135063318526315255121767} a^{3} + \frac{11714343822197714764820134425709557225327136599181297309328338}{490499861897398801256769367859796666844135063318526315255121767} a^{2} + \frac{73922569647631733496812481563014269899612155726582389210467447}{980999723794797602513538735719593333688270126637052630510243534} a - \frac{247739984376368673823231280186104774435323399627040304694087411}{980999723794797602513538735719593333688270126637052630510243534}$

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

Class group and class number

$C_{2}\times C_{10}$, which has order $20$ (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:  \( -1 \) (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:  \( 6648808169.565157 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.0.44217.1, 5.1.578000.2, 10.2.5679428000000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
17Data not computed