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Properties

Label 20.4.19290722540...8125.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{31}\cdot 23^{10}$
Root discriminant $58.11$
Ramified primes $5, 23$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2214298096, -1225863600, 2109281000, 298330310, -1286617875, -85381863, 303998195, -29312330, -27752675, 14891330, -3611196, -255555, 351970, -97645, 14180, 198, -945, 280, -15, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 15*x^18 + 280*x^17 - 945*x^16 + 198*x^15 + 14180*x^14 - 97645*x^13 + 351970*x^12 - 255555*x^11 - 3611196*x^10 + 14891330*x^9 - 27752675*x^8 - 29312330*x^7 + 303998195*x^6 - 85381863*x^5 - 1286617875*x^4 + 298330310*x^3 + 2109281000*x^2 - 1225863600*x + 2214298096)
 
gp: K = bnfinit(x^20 - 5*x^19 - 15*x^18 + 280*x^17 - 945*x^16 + 198*x^15 + 14180*x^14 - 97645*x^13 + 351970*x^12 - 255555*x^11 - 3611196*x^10 + 14891330*x^9 - 27752675*x^8 - 29312330*x^7 + 303998195*x^6 - 85381863*x^5 - 1286617875*x^4 + 298330310*x^3 + 2109281000*x^2 - 1225863600*x + 2214298096, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 15 x^{18} + 280 x^{17} - 945 x^{16} + 198 x^{15} + 14180 x^{14} - 97645 x^{13} + 351970 x^{12} - 255555 x^{11} - 3611196 x^{10} + 14891330 x^{9} - 27752675 x^{8} - 29312330 x^{7} + 303998195 x^{6} - 85381863 x^{5} - 1286617875 x^{4} + 298330310 x^{3} + 2109281000 x^{2} - 1225863600 x + 2214298096 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(192907225404162891209125518798828125=5^{31}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.11$
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:  $5, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{5064} a^{18} + \frac{223}{2532} a^{17} + \frac{97}{1688} a^{16} - \frac{73}{1688} a^{15} + \frac{125}{844} a^{14} - \frac{21}{422} a^{13} + \frac{61}{633} a^{12} - \frac{2093}{5064} a^{11} + \frac{1783}{5064} a^{10} + \frac{1105}{2532} a^{9} - \frac{159}{844} a^{8} + \frac{373}{1266} a^{7} + \frac{2345}{5064} a^{6} - \frac{1355}{5064} a^{5} + \frac{121}{2532} a^{4} + \frac{449}{1688} a^{3} - \frac{761}{2532} a^{2} - \frac{31}{633} a - \frac{224}{633}$, $\frac{1}{73990315546306241854360084687820472502673284498006655392733058226890839787132939784729224} a^{19} - \frac{7220674277029681821961425819346536713908510494242924101437303550022736736060290948721}{73990315546306241854360084687820472502673284498006655392733058226890839787132939784729224} a^{18} + \frac{1864983256134728941946782841297622524852468311106296623996073806120820406103038472371147}{24663438515435413951453361562606824167557761499335551797577686075630279929044313261576408} a^{17} + \frac{503480147100058881685179136559314868134877186480866539437821005584957727029924915862485}{6165859628858853487863340390651706041889440374833887949394421518907569982261078315394102} a^{16} - \frac{957329631189016051138090064600173077702052663556494315464541339890176906985945576257783}{24663438515435413951453361562606824167557761499335551797577686075630279929044313261576408} a^{15} - \frac{52505361760972127095496729752056944019278481003259685701407410839344070252079284098437}{649037855669352998722456883226495372830467407877251363094149633569217892869587191094116} a^{14} + \frac{2662956288596523067451560835137049824727289021825448700443748170984776165177572429518043}{18497578886576560463590021171955118125668321124501663848183264556722709946783234946182306} a^{13} - \frac{13561206748088086736542943066416521412430458376620205980853070585596561195136508351462025}{73990315546306241854360084687820472502673284498006655392733058226890839787132939784729224} a^{12} + \frac{9335505344147027713690228523974465369143807833019618023271395453706073268505975297921917}{36995157773153120927180042343910236251336642249003327696366529113445419893566469892364612} a^{11} - \frac{8795657089864340418591097399957776120782666516639051038086644821517868948648115688904679}{73990315546306241854360084687820472502673284498006655392733058226890839787132939784729224} a^{10} - \frac{1377359027780778770047477255012494787858335574964351424607895584967271105383822911738213}{6165859628858853487863340390651706041889440374833887949394421518907569982261078315394102} a^{9} - \frac{8504602737104988276335532400151500474965529902897029857696003552313555596293593248391119}{36995157773153120927180042343910236251336642249003327696366529113445419893566469892364612} a^{8} + \frac{28110204902457210372485581103397837421504690168159691711720824709683620112197112975968929}{73990315546306241854360084687820472502673284498006655392733058226890839787132939784729224} a^{7} + \frac{13373573806633773626564818180104060909629020594071735474477630493798739593445164004647919}{36995157773153120927180042343910236251336642249003327696366529113445419893566469892364612} a^{6} + \frac{632426538756151457830876930432168300513288173247272332781028232623673005474712953456031}{73990315546306241854360084687820472502673284498006655392733058226890839787132939784729224} a^{5} - \frac{11298086420236591553073927963342188150090569204944729793083667527534571582593429091519485}{24663438515435413951453361562606824167557761499335551797577686075630279929044313261576408} a^{4} - \frac{33983424521083692449271404981913258493271804059482098243345418846953859383258405641921639}{73990315546306241854360084687820472502673284498006655392733058226890839787132939784729224} a^{3} - \frac{6972302785016240852036563002584426882858148399284967073485836333841226251034969100923603}{36995157773153120927180042343910236251336642249003327696366529113445419893566469892364612} a^{2} - \frac{7083757706262346347389912468026739225727674426497348821176123191337025482153916110235361}{18497578886576560463590021171955118125668321124501663848183264556722709946783234946182306} a + \frac{770863671870831103812230283866736029533737912622162910855799973694422251672850644424819}{3082929814429426743931670195325853020944720187416943974697210759453784991130539157697051}$

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:  $11$
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

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.66125.1, 5.1.78125.1, 10.2.30517578125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
5Data not computed
$23$23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$