Properties

Label 20.0.23925952148...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 280001^{2}$
Root discriminant $11.72$
Ramified primes $5, 280001$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois Group 20T654

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 5, -7, 12, -19, 33, -56, 79, -104, 116, -121, 114, -97, 77, -53, 34, -18, 9, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 9*x^18 - 18*x^17 + 34*x^16 - 53*x^15 + 77*x^14 - 97*x^13 + 114*x^12 - 121*x^11 + 116*x^10 - 104*x^9 + 79*x^8 - 56*x^7 + 33*x^6 - 19*x^5 + 12*x^4 - 7*x^3 + 5*x^2 - 2*x + 1)
gp: K = bnfinit(x^20 - 3*x^19 + 9*x^18 - 18*x^17 + 34*x^16 - 53*x^15 + 77*x^14 - 97*x^13 + 114*x^12 - 121*x^11 + 116*x^10 - 104*x^9 + 79*x^8 - 56*x^7 + 33*x^6 - 19*x^5 + 12*x^4 - 7*x^3 + 5*x^2 - 2*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 3 x^{19} \) \(\mathstrut +\mathstrut 9 x^{18} \) \(\mathstrut -\mathstrut 18 x^{17} \) \(\mathstrut +\mathstrut 34 x^{16} \) \(\mathstrut -\mathstrut 53 x^{15} \) \(\mathstrut +\mathstrut 77 x^{14} \) \(\mathstrut -\mathstrut 97 x^{13} \) \(\mathstrut +\mathstrut 114 x^{12} \) \(\mathstrut -\mathstrut 121 x^{11} \) \(\mathstrut +\mathstrut 116 x^{10} \) \(\mathstrut -\mathstrut 104 x^{9} \) \(\mathstrut +\mathstrut 79 x^{8} \) \(\mathstrut -\mathstrut 56 x^{7} \) \(\mathstrut +\mathstrut 33 x^{6} \) \(\mathstrut -\mathstrut 19 x^{5} \) \(\mathstrut +\mathstrut 12 x^{4} \) \(\mathstrut -\mathstrut 7 x^{3} \) \(\mathstrut +\mathstrut 5 x^{2} \) \(\mathstrut -\mathstrut 2 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(2392595214874267578125=5^{15}\cdot 280001^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.72$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 280001$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{7} a^{18} + \frac{3}{7} a^{17} - \frac{1}{7} a^{15} + \frac{2}{7} a^{13} - \frac{2}{7} a^{12} - \frac{2}{7} a^{11} + \frac{2}{7} a^{10} + \frac{1}{7} a^{9} - \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{45787} a^{19} + \frac{1453}{45787} a^{18} - \frac{3707}{45787} a^{17} + \frac{387}{1477} a^{16} + \frac{22819}{45787} a^{15} - \frac{3869}{45787} a^{14} - \frac{198}{6541} a^{13} + \frac{16138}{45787} a^{12} - \frac{1616}{6541} a^{11} - \frac{216}{1477} a^{10} - \frac{22793}{45787} a^{9} - \frac{4219}{45787} a^{8} + \frac{2691}{6541} a^{7} - \frac{6338}{45787} a^{6} - \frac{18367}{45787} a^{5} - \frac{15845}{45787} a^{4} + \frac{6340}{45787} a^{3} - \frac{11400}{45787} a^{2} - \frac{554}{6541} a + \frac{18136}{45787}$

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

Class group and class number

Trivial Abelian group, 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:  $9$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( -\frac{19365}{45787} a^{19} + \frac{80529}{45787} a^{18} - \frac{197650}{45787} a^{17} + \frac{13336}{1477} a^{16} - \frac{699485}{45787} a^{15} + \frac{1068754}{45787} a^{14} - \frac{1430312}{45787} a^{13} + \frac{242521}{6541} a^{12} - \frac{1799985}{45787} a^{11} + \frac{56946}{1477} a^{10} - \frac{1543911}{45787} a^{9} + \frac{1089651}{45787} a^{8} - \frac{765314}{45787} a^{7} + \frac{248604}{45787} a^{6} - \frac{74953}{45787} a^{5} + \frac{65525}{45787} a^{4} - \frac{25694}{45787} a^{3} + \frac{93824}{45787} a^{2} + \frac{19872}{45787} a + \frac{15355}{45787} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{22434}{45787} a^{19} - \frac{23365}{45787} a^{18} + \frac{64846}{45787} a^{17} - \frac{1325}{1477} a^{16} + \frac{42409}{45787} a^{15} + \frac{60793}{45787} a^{14} - \frac{180758}{45787} a^{13} + \frac{407625}{45787} a^{12} - \frac{577839}{45787} a^{11} + \frac{3237}{211} a^{10} - \frac{924309}{45787} a^{9} + \frac{764521}{45787} a^{8} - \frac{835836}{45787} a^{7} + \frac{564092}{45787} a^{6} - \frac{302410}{45787} a^{5} + \frac{206686}{45787} a^{4} - \frac{41731}{45787} a^{3} + \frac{16671}{6541} a^{2} - \frac{23375}{45787} a + \frac{19365}{45787} \),  \( \frac{6862}{45787} a^{19} - \frac{63408}{45787} a^{18} + \frac{137876}{45787} a^{17} - \frac{11868}{1477} a^{16} + \frac{594210}{45787} a^{15} - \frac{1045719}{45787} a^{14} + \frac{1373492}{45787} a^{13} - \frac{1838195}{45787} a^{12} + \frac{1922157}{45787} a^{11} - \frac{9664}{211} a^{10} + \frac{1827765}{45787} a^{9} - \frac{1465496}{45787} a^{8} + \frac{1121304}{45787} a^{7} - \frac{471199}{45787} a^{6} + \frac{285438}{45787} a^{5} - \frac{75839}{45787} a^{4} + \frac{79381}{45787} a^{3} - \frac{18180}{6541} a^{2} + \frac{76444}{45787} a - \frac{39080}{45787} \),  \( \frac{72505}{45787} a^{19} - \frac{34514}{6541} a^{18} + \frac{660837}{45787} a^{17} - \frac{43444}{1477} a^{16} + \frac{2411602}{45787} a^{15} - \frac{3739430}{45787} a^{14} + \frac{5217171}{45787} a^{13} - \frac{6443980}{45787} a^{12} + \frac{7253399}{45787} a^{11} - \frac{243099}{1477} a^{10} + \frac{6936809}{45787} a^{9} - \frac{5889089}{45787} a^{8} + \frac{4365650}{45787} a^{7} - \frac{2693627}{45787} a^{6} + \frac{1565777}{45787} a^{5} - \frac{870061}{45787} a^{4} + \frac{85945}{6541} a^{3} - \frac{435241}{45787} a^{2} + \frac{180184}{45787} a - \frac{91206}{45787} \),  \( \frac{68686}{45787} a^{19} - \frac{224214}{45787} a^{18} + \frac{610818}{45787} a^{17} - \frac{39966}{1477} a^{16} + \frac{2189190}{45787} a^{15} - \frac{3386624}{45787} a^{14} + \frac{4656110}{45787} a^{13} - \frac{5719649}{45787} a^{12} + \frac{6308718}{45787} a^{11} - \frac{209656}{1477} a^{10} + \frac{825413}{6541} a^{9} - \frac{692456}{6541} a^{8} + \frac{3364292}{45787} a^{7} - \frac{271292}{6541} a^{6} + \frac{1040386}{45787} a^{5} - \frac{476337}{45787} a^{4} + \frac{414248}{45787} a^{3} - \frac{298176}{45787} a^{2} + \frac{135655}{45787} a - \frac{57236}{45787} \),  \( \frac{24053}{45787} a^{19} - \frac{10215}{6541} a^{18} + \frac{185589}{45787} a^{17} - \frac{11359}{1477} a^{16} + \frac{605328}{45787} a^{15} - \frac{891826}{45787} a^{14} + \frac{1199022}{45787} a^{13} - \frac{1401564}{45787} a^{12} + \frac{1522494}{45787} a^{11} - \frac{49158}{1477} a^{10} + \frac{1347873}{45787} a^{9} - \frac{1081798}{45787} a^{8} + \frac{782752}{45787} a^{7} - \frac{415451}{45787} a^{6} + \frac{298575}{45787} a^{5} - \frac{171945}{45787} a^{4} + \frac{26042}{6541} a^{3} - \frac{64349}{45787} a^{2} - \frac{2874}{45787} a + \frac{5918}{45787} \),  \( \frac{50161}{45787} a^{19} - \frac{100545}{45787} a^{18} + \frac{314689}{45787} a^{17} - \frac{16151}{1477} a^{16} + \frac{956173}{45787} a^{15} - \frac{1263852}{45787} a^{14} + \frac{259000}{6541} a^{13} - \frac{1938996}{45787} a^{12} + \frac{322946}{6541} a^{11} - \frac{67446}{1477} a^{10} + \frac{1858984}{45787} a^{9} - \frac{1558503}{45787} a^{8} + \frac{133995}{6541} a^{7} - \frac{708082}{45787} a^{6} + \frac{339436}{45787} a^{5} - \frac{305021}{45787} a^{4} + \frac{213173}{45787} a^{3} - \frac{93131}{45787} a^{2} + \frac{10056}{6541} a - \frac{22007}{45787} \),  \( \frac{41851}{45787} a^{19} - \frac{139535}{45787} a^{18} + \frac{377159}{45787} a^{17} - \frac{25554}{1477} a^{16} + \frac{1398661}{45787} a^{15} - \frac{2216463}{45787} a^{14} + \frac{3061290}{45787} a^{13} - \frac{549375}{6541} a^{12} + \frac{4290341}{45787} a^{11} - \frac{145744}{1477} a^{10} + \frac{4085017}{45787} a^{9} - \frac{3527214}{45787} a^{8} + \frac{2616203}{45787} a^{7} - \frac{1538141}{45787} a^{6} + \frac{995612}{45787} a^{5} - \frac{453844}{45787} a^{4} + \frac{346348}{45787} a^{3} - \frac{288664}{45787} a^{2} + \frac{147557}{45787} a - \frac{86396}{45787} \),  \( \frac{1507}{6541} a^{19} - \frac{37112}{45787} a^{18} + \frac{147398}{45787} a^{17} - \frac{1472}{211} a^{16} + \frac{682554}{45787} a^{15} - \frac{159536}{6541} a^{14} + \frac{1764291}{45787} a^{13} - \frac{2324900}{45787} a^{12} + \frac{2927543}{45787} a^{11} - \frac{101697}{1477} a^{10} + \frac{3254680}{45787} a^{9} - \frac{2970881}{45787} a^{8} + \frac{2409562}{45787} a^{7} - \frac{1783153}{45787} a^{6} + \frac{1044038}{45787} a^{5} - \frac{82257}{6541} a^{4} + \frac{273657}{45787} a^{3} - \frac{257414}{45787} a^{2} + \frac{135753}{45787} a - \frac{92619}{45787} \),  \( \frac{9505}{45787} a^{19} + \frac{3191}{6541} a^{18} - \frac{44455}{45787} a^{17} + \frac{6613}{1477} a^{16} - \frac{358179}{45787} a^{15} + \frac{724708}{45787} a^{14} - \frac{1053457}{45787} a^{13} + \frac{1529293}{45787} a^{12} - \frac{1785295}{45787} a^{11} + \frac{64516}{1477} a^{10} - \frac{2050337}{45787} a^{9} + \frac{1715018}{45787} a^{8} - \frac{1564407}{45787} a^{7} + \frac{863332}{45787} a^{6} - \frac{594276}{45787} a^{5} + \frac{261440}{45787} a^{4} - \frac{15027}{6541} a^{3} + \frac{236682}{45787} a^{2} - \frac{8396}{45787} a + \frac{92740}{45787} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 858.376938924 \) (assuming GRH)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

20T654:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 57600
The 70 conjugacy class representatives for t20n654 are not computed
Character table for t20n654 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.2.875003125.1

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 $20$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
5Data not computed
280001Data not computed