Properties

Label 20.0.26150594311...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 199^{2}\cdot 1471^{2}$
Root discriminant $11.77$
Ramified primes $5, 199, 1471$
Class number $1$
Class group Trivial
Galois Group 20T654

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 6, -6, 10, -10, 27, -48, 88, -112, 135, -128, 118, -93, 73, -50, 33, -18, 9, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 9*x^18 - 18*x^17 + 33*x^16 - 50*x^15 + 73*x^14 - 93*x^13 + 118*x^12 - 128*x^11 + 135*x^10 - 112*x^9 + 88*x^8 - 48*x^7 + 27*x^6 - 10*x^5 + 10*x^4 - 6*x^3 + 6*x^2 - 2*x + 1)
gp: K = bnfinit(x^20 - 3*x^19 + 9*x^18 - 18*x^17 + 33*x^16 - 50*x^15 + 73*x^14 - 93*x^13 + 118*x^12 - 128*x^11 + 135*x^10 - 112*x^9 + 88*x^8 - 48*x^7 + 27*x^6 - 10*x^5 + 10*x^4 - 6*x^3 + 6*x^2 - 2*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 3 x^{19} \) \(\mathstrut +\mathstrut 9 x^{18} \) \(\mathstrut -\mathstrut 18 x^{17} \) \(\mathstrut +\mathstrut 33 x^{16} \) \(\mathstrut -\mathstrut 50 x^{15} \) \(\mathstrut +\mathstrut 73 x^{14} \) \(\mathstrut -\mathstrut 93 x^{13} \) \(\mathstrut +\mathstrut 118 x^{12} \) \(\mathstrut -\mathstrut 128 x^{11} \) \(\mathstrut +\mathstrut 135 x^{10} \) \(\mathstrut -\mathstrut 112 x^{9} \) \(\mathstrut +\mathstrut 88 x^{8} \) \(\mathstrut -\mathstrut 48 x^{7} \) \(\mathstrut +\mathstrut 27 x^{6} \) \(\mathstrut -\mathstrut 10 x^{5} \) \(\mathstrut +\mathstrut 10 x^{4} \) \(\mathstrut -\mathstrut 6 x^{3} \) \(\mathstrut +\mathstrut 6 x^{2} \) \(\mathstrut -\mathstrut 2 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(2615059431182861328125=5^{15}\cdot 199^{2}\cdot 1471^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.77$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 199, 1471$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{287389} a^{19} + \frac{98926}{287389} a^{18} - \frac{94743}{287389} a^{17} + \frac{74581}{287389} a^{16} + \frac{85985}{287389} a^{15} - \frac{16996}{287389} a^{14} + \frac{115828}{287389} a^{13} - \frac{26089}{287389} a^{12} + \frac{82046}{287389} a^{11} + \frac{1079}{287389} a^{10} + \frac{123207}{287389} a^{9} + \frac{2923}{287389} a^{8} + \frac{56221}{287389} a^{7} + \frac{47944}{287389} a^{6} - \frac{16053}{287389} a^{5} + \frac{4367}{287389} a^{4} + \frac{77286}{287389} a^{3} + \frac{129732}{287389} a^{2} + \frac{39072}{287389} a - \frac{28164}{287389}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( -\frac{1248594}{287389} a^{19} + \frac{3968657}{287389} a^{18} - \frac{11301787}{287389} a^{17} + \frac{22822731}{287389} a^{16} - \frac{40393431}{287389} a^{15} + \frac{60651554}{287389} a^{14} - \frac{86170840}{287389} a^{13} + \frac{109095703}{287389} a^{12} - \frac{134820603}{287389} a^{11} + \frac{145465540}{287389} a^{10} - \frac{146206316}{287389} a^{9} + \frac{117742139}{287389} a^{8} - \frac{81759388}{287389} a^{7} + \frac{40772424}{287389} a^{6} - \frac{16647496}{287389} a^{5} + \frac{7493613}{287389} a^{4} - \frac{8841301}{287389} a^{3} + \frac{7661710}{287389} a^{2} - \frac{4518075}{287389} a + \frac{1632932}{287389} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{1529931}{287389} a^{19} - \frac{4037522}{287389} a^{18} + \frac{11632757}{287389} a^{17} - \frac{21549268}{287389} a^{16} + \frac{37599800}{287389} a^{15} - \frac{53492299}{287389} a^{14} + \frac{76062940}{287389} a^{13} - \frac{91738296}{287389} a^{12} + \frac{114769173}{287389} a^{11} - \frac{115209856}{287389} a^{10} + \frac{116156162}{287389} a^{9} - \frac{80828225}{287389} a^{8} + \frac{57063018}{287389} a^{7} - \frac{19974614}{287389} a^{6} + \frac{11777157}{287389} a^{5} - \frac{4034241}{287389} a^{4} + \frac{9837888}{287389} a^{3} - \frac{4204358}{287389} a^{2} + \frac{3425922}{287389} a - \frac{169136}{287389} \),  \( \frac{19450}{287389} a^{19} + \frac{616123}{287389} a^{18} - \frac{1737416}{287389} a^{17} + \frac{5033780}{287389} a^{16} - \frac{9679567}{287389} a^{15} + \frac{16881101}{287389} a^{14} - \frac{24415836}{287389} a^{13} + \frac{34584604}{287389} a^{12} - \frac{42322600}{287389} a^{11} + \frac{52599340}{287389} a^{10} - \frac{54189853}{287389} a^{9} + \frac{53978460}{287389} a^{8} - \frac{39388988}{287389} a^{7} + \frac{26373283}{287389} a^{6} - \frac{10472400}{287389} a^{5} + \frac{4756619}{287389} a^{4} - \frac{2418271}{287389} a^{3} + \frac{4610204}{287389} a^{2} - \frac{2492617}{287389} a + \frac{1410579}{287389} \),  \( \frac{596514}{287389} a^{19} - \frac{2288074}{287389} a^{18} + \frac{6418284}{287389} a^{17} - \frac{13863405}{287389} a^{16} + \frac{24794747}{287389} a^{15} - \frac{38640317}{287389} a^{14} + \frac{55001067}{287389} a^{13} - \frac{71899257}{287389} a^{12} + \frac{88331534}{287389} a^{11} - \frac{99549348}{287389} a^{10} + \frac{100148022}{287389} a^{9} - \frac{87057508}{287389} a^{8} + \frac{60968096}{287389} a^{7} - \frac{35310577}{287389} a^{6} + \frac{13756910}{287389} a^{5} - \frac{6814594}{287389} a^{4} + \frac{4985404}{287389} a^{3} - \frac{6041285}{287389} a^{2} + \frac{3195776}{287389} a - \frac{1758468}{287389} \),  \( \frac{1226}{4871} a^{19} - \frac{28979}{4871} a^{18} + \frac{76884}{4871} a^{17} - \frac{211559}{4871} a^{16} + \frac{389108}{4871} a^{15} - \frac{666285}{4871} a^{14} + \frac{945839}{4871} a^{13} - \frac{1327040}{4871} a^{12} + \frac{1599934}{4871} a^{11} - \frac{1974813}{4871} a^{10} + \frac{1974827}{4871} a^{9} - \frac{1949858}{4871} a^{8} + \frac{1341821}{4871} a^{7} - \frac{909890}{4871} a^{6} + \frac{309606}{4871} a^{5} - \frac{169772}{4871} a^{4} + \frac{75009}{4871} a^{3} - \frac{162074}{4871} a^{2} + \frac{69052}{4871} a - \frac{52126}{4871} \),  \( \frac{2348837}{287389} a^{19} - \frac{5602554}{287389} a^{18} + \frac{16165986}{287389} a^{17} - \frac{28378942}{287389} a^{16} + \frac{48686713}{287389} a^{15} - \frac{66589299}{287389} a^{14} + \frac{94247943}{287389} a^{13} - \frac{109209399}{287389} a^{12} + \frac{137335048}{287389} a^{11} - \frac{129126129}{287389} a^{10} + \frac{128947034}{287389} a^{9} - \frac{75655966}{287389} a^{8} + \frac{51886442}{287389} a^{7} - \frac{7348469}{287389} a^{6} + \frac{8178109}{287389} a^{5} - \frac{2128732}{287389} a^{4} + \frac{12725758}{287389} a^{3} - \frac{2396295}{287389} a^{2} + \frac{2979450}{287389} a + \frac{1141253}{287389} \),  \( \frac{122583}{287389} a^{19} + \frac{267003}{287389} a^{18} - \frac{779068}{287389} a^{17} + \frac{3105134}{287389} a^{16} - \frac{6302267}{287389} a^{15} + \frac{11932531}{287389} a^{14} - \frac{17440550}{287389} a^{13} + \frac{25861915}{287389} a^{12} - \frac{31633416}{287389} a^{11} + \frac{41164744}{287389} a^{10} - \frac{42891397}{287389} a^{9} + \frac{45343488}{287389} a^{8} - \frac{33473890}{287389} a^{7} + \frac{23867589}{287389} a^{6} - \frac{8981475}{287389} a^{5} + \frac{3937700}{287389} a^{4} - \frac{978203}{287389} a^{3} + \frac{3716109}{287389} a^{2} - \frac{2073821}{287389} a + \frac{1413390}{287389} \),  \( \frac{93461}{287389} a^{19} - \frac{443411}{287389} a^{18} + \frac{1116512}{287389} a^{17} - \frac{2504466}{287389} a^{16} + \frac{4296313}{287389} a^{15} - \frac{6674100}{287389} a^{14} + \frac{9228304}{287389} a^{13} - \frac{12166091}{287389} a^{12} + \frac{14357358}{287389} a^{11} - \frac{16410293}{287389} a^{10} + \frac{15465981}{287389} a^{9} - \frac{13340330}{287389} a^{8} + \frac{7897297}{287389} a^{7} - \frac{4385939}{287389} a^{6} + \frac{703314}{287389} a^{5} - \frac{235582}{287389} a^{4} + \frac{279109}{287389} a^{3} - \frac{634236}{287389} a^{2} + \frac{143558}{287389} a - \frac{327142}{287389} \),  \( \frac{19169}{287389} a^{19} + \frac{407261}{287389} a^{18} - \frac{979643}{287389} a^{17} + \frac{3044193}{287389} a^{16} - \frac{5389841}{287389} a^{15} + \frac{9586639}{287389} a^{14} - \frac{13280376}{287389} a^{13} + \frac{19211882}{287389} a^{12} - \frac{22556565}{287389} a^{11} + \frac{29017632}{287389} a^{10} - \frac{27884552}{287389} a^{9} + \frac{29016421}{287389} a^{8} - \frac{18401297}{287389} a^{7} + \frac{14050575}{287389} a^{6} - \frac{3662395}{287389} a^{5} + \frac{3242103}{287389} a^{4} - \frac{569739}{287389} a^{3} + \frac{2354803}{287389} a^{2} - \frac{539344}{287389} a + \frac{990382}{287389} \),  \( \frac{659985}{287389} a^{19} - \frac{2805578}{287389} a^{18} + \frac{7749712}{287389} a^{17} - \frac{17165641}{287389} a^{16} + \frac{30579352}{287389} a^{15} - \frac{48018964}{287389} a^{14} + \frac{67954551}{287389} a^{13} - \frac{89389487}{287389} a^{12} + \frac{109076528}{287389} a^{11} - \frac{124178175}{287389} a^{10} + \frac{123643338}{287389} a^{9} - \frac{108739244}{287389} a^{8} + \frac{74081868}{287389} a^{7} - \frac{42891188}{287389} a^{6} + \frac{15662675}{287389} a^{5} - \frac{7829289}{287389} a^{4} + \frac{6399214}{287389} a^{3} - \frac{7815475}{287389} a^{2} + \frac{4117174}{287389} a - \frac{2083521}{287389} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 906.492535524 \)
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.2.914778125.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ $20$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$199$199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.4.2.1$x^{4} + 2189 x^{2} + 1425636$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
199.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$
1471Data not computed