Properties

Label 20.0.70900030741...9856.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{4}\cdot 83^{4}\cdot 983^{4}$
Root discriminant $11.03$
Ramified primes $2, 83, 983$
Class number $1$
Class group Trivial
Galois Group 20T669

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 15, -37, 78, -139, 219, -307, 386, -442, 463, -442, 386, -307, 219, -139, 78, -37, 15, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 15*x^18 - 37*x^17 + 78*x^16 - 139*x^15 + 219*x^14 - 307*x^13 + 386*x^12 - 442*x^11 + 463*x^10 - 442*x^9 + 386*x^8 - 307*x^7 + 219*x^6 - 139*x^5 + 78*x^4 - 37*x^3 + 15*x^2 - 5*x + 1)
gp: K = bnfinit(x^20 - 5*x^19 + 15*x^18 - 37*x^17 + 78*x^16 - 139*x^15 + 219*x^14 - 307*x^13 + 386*x^12 - 442*x^11 + 463*x^10 - 442*x^9 + 386*x^8 - 307*x^7 + 219*x^6 - 139*x^5 + 78*x^4 - 37*x^3 + 15*x^2 - 5*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 5 x^{19} \) \(\mathstrut +\mathstrut 15 x^{18} \) \(\mathstrut -\mathstrut 37 x^{17} \) \(\mathstrut +\mathstrut 78 x^{16} \) \(\mathstrut -\mathstrut 139 x^{15} \) \(\mathstrut +\mathstrut 219 x^{14} \) \(\mathstrut -\mathstrut 307 x^{13} \) \(\mathstrut +\mathstrut 386 x^{12} \) \(\mathstrut -\mathstrut 442 x^{11} \) \(\mathstrut +\mathstrut 463 x^{10} \) \(\mathstrut -\mathstrut 442 x^{9} \) \(\mathstrut +\mathstrut 386 x^{8} \) \(\mathstrut -\mathstrut 307 x^{7} \) \(\mathstrut +\mathstrut 219 x^{6} \) \(\mathstrut -\mathstrut 139 x^{5} \) \(\mathstrut +\mathstrut 78 x^{4} \) \(\mathstrut -\mathstrut 37 x^{3} \) \(\mathstrut +\mathstrut 15 x^{2} \) \(\mathstrut -\mathstrut 5 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(709000307415298179856=2^{4}\cdot 83^{4}\cdot 983^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.03$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 83, 983$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{62} a^{18} - \frac{9}{31} a^{17} - \frac{19}{62} a^{15} + \frac{15}{62} a^{14} - \frac{5}{62} a^{13} + \frac{21}{62} a^{12} - \frac{17}{62} a^{11} + \frac{14}{31} a^{10} + \frac{17}{62} a^{9} + \frac{14}{31} a^{8} - \frac{17}{62} a^{7} + \frac{21}{62} a^{6} - \frac{5}{62} a^{5} + \frac{15}{62} a^{4} - \frac{19}{62} a^{3} - \frac{9}{31} a + \frac{1}{62}$, $\frac{1}{3782} a^{19} - \frac{3}{1891} a^{18} - \frac{325}{1891} a^{17} + \frac{1345}{3782} a^{16} - \frac{1267}{3782} a^{15} - \frac{1251}{3782} a^{14} - \frac{1031}{3782} a^{13} + \frac{297}{3782} a^{12} + \frac{563}{1891} a^{11} - \frac{1507}{3782} a^{10} - \frac{845}{1891} a^{9} + \frac{1187}{3782} a^{8} - \frac{679}{3782} a^{7} + \frac{433}{3782} a^{6} + \frac{823}{3782} a^{5} - \frac{1389}{3782} a^{4} - \frac{517}{1891} a^{3} - \frac{691}{1891} a^{2} + \frac{1397}{3782} a - \frac{335}{1891}$

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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{2634}{1891} a^{19} - \frac{12571}{1891} a^{18} + \frac{37502}{1891} a^{17} - \frac{93663}{1891} a^{16} + \frac{197977}{1891} a^{15} - \frac{353409}{1891} a^{14} + \frac{562303}{1891} a^{13} - \frac{793088}{1891} a^{12} + \frac{1001013}{1891} a^{11} - \frac{1152092}{1891} a^{10} + \frac{1206534}{1891} a^{9} - \frac{1149237}{1891} a^{8} + \frac{1002508}{1891} a^{7} - \frac{792263}{1891} a^{6} + \frac{17923}{61} a^{5} - \frac{348156}{1891} a^{4} + \frac{187669}{1891} a^{3} - \frac{81326}{1891} a^{2} + \frac{30495}{1891} a - \frac{8590}{1891} \),  \( \frac{599}{1891} a^{19} - \frac{2008}{1891} a^{18} + \frac{3795}{1891} a^{17} - \frac{7475}{1891} a^{16} + \frac{12717}{1891} a^{15} - \frac{14543}{1891} a^{14} + \frac{15550}{1891} a^{13} - \frac{15711}{1891} a^{12} + \frac{12136}{1891} a^{11} - \frac{13008}{1891} a^{10} + \frac{14991}{1891} a^{9} - \frac{14216}{1891} a^{8} + \frac{16375}{1891} a^{7} - \frac{21233}{1891} a^{6} + \frac{17970}{1891} a^{5} - \frac{17783}{1891} a^{4} + \frac{16132}{1891} a^{3} - \frac{9015}{1891} a^{2} + \frac{4580}{1891} a - \frac{2634}{1891} \),  \( \frac{5240}{1891} a^{19} - \frac{24730}{1891} a^{18} + \frac{71793}{1891} a^{17} - \frac{173929}{1891} a^{16} + \frac{358718}{1891} a^{15} - \frac{622746}{1891} a^{14} + \frac{959372}{1891} a^{13} - \frac{1307609}{1891} a^{12} + \frac{1595714}{1891} a^{11} - \frac{1774842}{1891} a^{10} + \frac{1797013}{1891} a^{9} - \frac{1653573}{1891} a^{8} + \frac{1388286}{1891} a^{7} - \frac{1050700}{1891} a^{6} + \frac{703089}{1891} a^{5} - \frac{413603}{1891} a^{4} + \frac{208663}{1891} a^{3} - \frac{84245}{1891} a^{2} + \frac{30719}{1891} a - \frac{7631}{1891} \),  \( \frac{1183}{3782} a^{19} - \frac{2523}{3782} a^{18} + \frac{1716}{1891} a^{17} - \frac{4869}{3782} a^{16} - \frac{564}{1891} a^{15} + \frac{12926}{1891} a^{14} - \frac{31282}{1891} a^{13} + \frac{64869}{1891} a^{12} - \frac{213133}{3782} a^{11} + \frac{289267}{3782} a^{10} - \frac{355747}{3782} a^{9} + \frac{390157}{3782} a^{8} - \frac{187122}{1891} a^{7} + \frac{168005}{1891} a^{6} - \frac{133535}{1891} a^{5} + \frac{90143}{1891} a^{4} - \frac{107471}{3782} a^{3} + \frac{27824}{1891} a^{2} - \frac{18139}{3782} a + \frac{6185}{3782} \),  \( \frac{3301}{3782} a^{19} - \frac{18037}{3782} a^{18} + \frac{26944}{1891} a^{17} - \frac{132593}{3782} a^{16} + \frac{140410}{1891} a^{15} - \frac{249385}{1891} a^{14} + \frac{389140}{1891} a^{13} - \frac{544514}{1891} a^{12} + \frac{1349577}{3782} a^{11} - \frac{1528839}{3782} a^{10} + \frac{1584235}{3782} a^{9} - \frac{1485825}{3782} a^{8} + \frac{638032}{1891} a^{7} - \frac{497800}{1891} a^{6} + \frac{342254}{1891} a^{5} - \frac{208632}{1891} a^{4} + \frac{225477}{3782} a^{3} - \frac{47720}{1891} a^{2} + \frac{33691}{3782} a - \frac{12559}{3782} \),  \( \frac{936}{1891} a^{19} - \frac{7145}{3782} a^{18} + \frac{10994}{1891} a^{17} - \frac{26960}{1891} a^{16} + \frac{107155}{3782} a^{15} - \frac{185339}{3782} a^{14} + \frac{288477}{3782} a^{13} - \frac{375547}{3782} a^{12} + \frac{453735}{3782} a^{11} - \frac{245208}{1891} a^{10} + \frac{472225}{3782} a^{9} - \frac{210289}{1891} a^{8} + \frac{334859}{3782} a^{7} - \frac{226851}{3782} a^{6} + \frac{136009}{3782} a^{5} - \frac{69257}{3782} a^{4} + \frac{17633}{3782} a^{3} - \frac{108}{1891} a^{2} + \frac{57}{1891} a + \frac{5471}{3782} \),  \( \frac{197}{62} a^{19} - \frac{461}{31} a^{18} + \frac{1308}{31} a^{17} - \frac{6285}{62} a^{16} + \frac{12929}{62} a^{15} - \frac{22261}{62} a^{14} + \frac{34107}{62} a^{13} - \frac{46515}{62} a^{12} + \frac{28349}{31} a^{11} - \frac{63237}{62} a^{10} + \frac{32223}{31} a^{9} - \frac{59581}{62} a^{8} + \frac{50483}{62} a^{7} - \frac{38757}{62} a^{6} + \frac{26415}{62} a^{5} - \frac{15905}{62} a^{4} + \frac{4243}{31} a^{3} - \frac{1835}{31} a^{2} + \frac{1387}{62} a - \frac{207}{31} \),  \( \frac{4413}{1891} a^{19} - \frac{45575}{3782} a^{18} + \frac{66138}{1891} a^{17} - \frac{159208}{1891} a^{16} + \frac{21367}{122} a^{15} - \frac{1154141}{3782} a^{14} + \frac{1766995}{3782} a^{13} - \frac{2423927}{3782} a^{12} + \frac{2955833}{3782} a^{11} - \frac{1645585}{1891} a^{10} + \frac{3351791}{3782} a^{9} - \frac{1545457}{1891} a^{8} + \frac{2595397}{3782} a^{7} - \frac{1983773}{3782} a^{6} + \frac{1334537}{3782} a^{5} - \frac{787471}{3782} a^{4} + \frac{408033}{3782} a^{3} - \frac{85386}{1891} a^{2} + \frac{28422}{1891} a - \frac{21247}{3782} \),  \( \frac{7059}{3782} a^{19} - \frac{13979}{1891} a^{18} + \frac{36452}{1891} a^{17} - \frac{168655}{3782} a^{16} + \frac{332273}{3782} a^{15} - \frac{17429}{122} a^{14} + \frac{800401}{3782} a^{13} - \frac{1046563}{3782} a^{12} + \frac{618241}{1891} a^{11} - \frac{1343343}{3782} a^{10} + \frac{668224}{1891} a^{9} - \frac{1198577}{3782} a^{8} + \frac{994497}{3782} a^{7} - \frac{744607}{3782} a^{6} + \frac{491943}{3782} a^{5} - \frac{292855}{3782} a^{4} + \frac{77048}{1891} a^{3} - \frac{31136}{1891} a^{2} + \frac{26271}{3782} a - \frac{1381}{1891} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 103.364702597 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

20T669:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 61440
The 126 conjugacy class representatives for t20n669 are not computed
Character table for t20n669 is not computed

Intermediate fields

5.5.81589.1, 10.2.6656764921.1, 10.4.26627059684.1, 10.4.26627059684.2

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$V_4$$[2]^{2}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
83Data not computed
983Data not computed