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Properties

Label 20.0.64553137783...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 7^{15}\cdot 11^{9}$
Root discriminant $49.03$
Ramified primes $3, 5, 7, 11$
Class number Not computed
Class group Not computed
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![87786948061, -45579456448, 15771143512, -3757533269, 3839375256, -323709400, 30457244, -10310764, 12205932, 3012761, 607603, -34407, 91673, -7861, -465, -2188, -193, 58, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 58*x^17 - 193*x^16 - 2188*x^15 - 465*x^14 - 7861*x^13 + 91673*x^12 - 34407*x^11 + 607603*x^10 + 3012761*x^9 + 12205932*x^8 - 10310764*x^7 + 30457244*x^6 - 323709400*x^5 + 3839375256*x^4 - 3757533269*x^3 + 15771143512*x^2 - 45579456448*x + 87786948061)
 
gp: K = bnfinit(x^20 - 2*x^19 + 58*x^17 - 193*x^16 - 2188*x^15 - 465*x^14 - 7861*x^13 + 91673*x^12 - 34407*x^11 + 607603*x^10 + 3012761*x^9 + 12205932*x^8 - 10310764*x^7 + 30457244*x^6 - 323709400*x^5 + 3839375256*x^4 - 3757533269*x^3 + 15771143512*x^2 - 45579456448*x + 87786948061, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 58 x^{17} - 193 x^{16} - 2188 x^{15} - 465 x^{14} - 7861 x^{13} + 91673 x^{12} - 34407 x^{11} + 607603 x^{10} + 3012761 x^{9} + 12205932 x^{8} - 10310764 x^{7} + 30457244 x^{6} - 323709400 x^{5} + 3839375256 x^{4} - 3757533269 x^{3} + 15771143512 x^{2} - 45579456448 x + 87786948061 \)

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:  \(6455313778301718076702695673828125=3^{10}\cdot 5^{10}\cdot 7^{15}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.03$
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:  $3, 5, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{16} a^{17} - \frac{1}{16} a^{16} + \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{2} a^{12} + \frac{7}{16} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{3}{8} a^{8} - \frac{3}{16} a^{7} - \frac{3}{8} a^{6} - \frac{7}{16} a^{5} + \frac{3}{8} a^{4} - \frac{3}{16} a^{3} - \frac{1}{8} a^{2} - \frac{7}{16} a + \frac{1}{16}$, $\frac{1}{64} a^{18} + \frac{3}{64} a^{16} + \frac{1}{8} a^{14} + \frac{1}{16} a^{13} + \frac{15}{64} a^{12} - \frac{11}{64} a^{11} - \frac{1}{4} a^{10} + \frac{3}{8} a^{9} - \frac{13}{64} a^{8} + \frac{7}{64} a^{7} + \frac{19}{64} a^{6} + \frac{15}{64} a^{5} - \frac{13}{64} a^{4} - \frac{5}{64} a^{3} - \frac{25}{64} a^{2} - \frac{11}{32} a - \frac{31}{64}$, $\frac{1}{3073268404094799150087733000034034575405390800286177162527877378969562659435463339964701477367234719488} a^{19} + \frac{8251445268177163805789869704683211324053250289229263544309977117602313066256958086516156774090345249}{3073268404094799150087733000034034575405390800286177162527877378969562659435463339964701477367234719488} a^{18} + \frac{75810035006116260279102744462160138335344392548089826141355593757052167220031861065962244098615136275}{3073268404094799150087733000034034575405390800286177162527877378969562659435463339964701477367234719488} a^{17} - \frac{299560684625322685886761787973620396930850782994018332306948247022866057236394913461776244417814337261}{3073268404094799150087733000034034575405390800286177162527877378969562659435463339964701477367234719488} a^{16} + \frac{101134701918164018854787079435502740049988546644148168341741515241476055343390786141782391851536731209}{384158550511849893760966625004254321925673850035772145315984672371195332429432917495587684670904339936} a^{15} + \frac{266427295686690854270517260124614312940224834304456202979836661967234657769701054593704523683614706515}{768317101023699787521933250008508643851347700071544290631969344742390664858865834991175369341808679872} a^{14} + \frac{640022258046195525135791398444467265873153881893772887573376878701740560988357646127430365708368727827}{3073268404094799150087733000034034575405390800286177162527877378969562659435463339964701477367234719488} a^{13} - \frac{77010872922811507346802213020200829267458762286917755989232358227057735478059865926448451621684084343}{768317101023699787521933250008508643851347700071544290631969344742390664858865834991175369341808679872} a^{12} + \frac{866688105715530079234521206532618274499876628819548853614843021942743669776681514640325394145753516917}{3073268404094799150087733000034034575405390800286177162527877378969562659435463339964701477367234719488} a^{11} + \frac{79659333823277827853973977092750420681751724260075858720645640288833564843459765407298359541446337541}{384158550511849893760966625004254321925673850035772145315984672371195332429432917495587684670904339936} a^{10} + \frac{891351428158526739976048832007286297054058627973348385949872061344178195286412546303730877235661299307}{3073268404094799150087733000034034575405390800286177162527877378969562659435463339964701477367234719488} a^{9} - \frac{487151987130103380837268496966134537826734822449776718730794849171496793655087679824229938833730875619}{1536634202047399575043866500017017287702695400143088581263938689484781329717731669982350738683617359744} a^{8} + \frac{142057813748516012994520911536014102609163222765509576637587443641372433060952644332202670328370484933}{1536634202047399575043866500017017287702695400143088581263938689484781329717731669982350738683617359744} a^{7} + \frac{319343928657089475934338177393186752713243479878957046455891397423166766363553931475332311211006988497}{1536634202047399575043866500017017287702695400143088581263938689484781329717731669982350738683617359744} a^{6} - \frac{477204413656221282856629370735629924377026416377706890038275973917888764943409816978135910405467902855}{1536634202047399575043866500017017287702695400143088581263938689484781329717731669982350738683617359744} a^{5} + \frac{235956735578688719175186602648913304501599914364002963279925802309954227267372853507387219519845720439}{1536634202047399575043866500017017287702695400143088581263938689484781329717731669982350738683617359744} a^{4} + \frac{173480460691689073524391874591119686650075460282140257438402482920376920483893402264987451975011131817}{1536634202047399575043866500017017287702695400143088581263938689484781329717731669982350738683617359744} a^{3} - \frac{7642821403670085942455394045301337159609445947101912713709540374934741879200679192257674984507158191}{3073268404094799150087733000034034575405390800286177162527877378969562659435463339964701477367234719488} a^{2} + \frac{4247895832063463315074059618786262704199624125431251256903719167007652532217636469167427690607067341}{43285470480208438733630042254000486977540715496988410739829258858726234639936103379784527850242742528} a - \frac{141501029148180295266868910605762958964015921033909862137605633098395955329566517614118946814046686703}{3073268404094799150087733000034034575405390800286177162527877378969562659435463339964701477367234719488}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.848925.1, 10.0.246071287.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ R R R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ $20$

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
3Data not computed
5Data not computed
7Data not computed
$11$11.10.9.5$x^{10} - 8019$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$