Properties

Label 20.0.36913968719...0048.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{10}\cdot 157^{5}$
Root discriminant $10.67$
Ramified primes $2, 3, 157$
Class number $1$
Class group Trivial
Galois Group $D_5\wr C_2$ (as 20T48)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 46, -129, 249, -344, 328, -158, -118, 372, -478, 398, -199, -4, 128, -153, 116, -64, 26, -7, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 26*x^18 - 64*x^17 + 116*x^16 - 153*x^15 + 128*x^14 - 4*x^13 - 199*x^12 + 398*x^11 - 478*x^10 + 372*x^9 - 118*x^8 - 158*x^7 + 328*x^6 - 344*x^5 + 249*x^4 - 129*x^3 + 46*x^2 - 10*x + 1)
gp: K = bnfinit(x^20 - 7*x^19 + 26*x^18 - 64*x^17 + 116*x^16 - 153*x^15 + 128*x^14 - 4*x^13 - 199*x^12 + 398*x^11 - 478*x^10 + 372*x^9 - 118*x^8 - 158*x^7 + 328*x^6 - 344*x^5 + 249*x^4 - 129*x^3 + 46*x^2 - 10*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 7 x^{19} \) \(\mathstrut +\mathstrut 26 x^{18} \) \(\mathstrut -\mathstrut 64 x^{17} \) \(\mathstrut +\mathstrut 116 x^{16} \) \(\mathstrut -\mathstrut 153 x^{15} \) \(\mathstrut +\mathstrut 128 x^{14} \) \(\mathstrut -\mathstrut 4 x^{13} \) \(\mathstrut -\mathstrut 199 x^{12} \) \(\mathstrut +\mathstrut 398 x^{11} \) \(\mathstrut -\mathstrut 478 x^{10} \) \(\mathstrut +\mathstrut 372 x^{9} \) \(\mathstrut -\mathstrut 118 x^{8} \) \(\mathstrut -\mathstrut 158 x^{7} \) \(\mathstrut +\mathstrut 328 x^{6} \) \(\mathstrut -\mathstrut 344 x^{5} \) \(\mathstrut +\mathstrut 249 x^{4} \) \(\mathstrut -\mathstrut 129 x^{3} \) \(\mathstrut +\mathstrut 46 x^{2} \) \(\mathstrut -\mathstrut 10 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(369139687194512130048=2^{16}\cdot 3^{10}\cdot 157^{5}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.67$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{31} a^{18} - \frac{1}{31} a^{17} - \frac{5}{31} a^{16} - \frac{7}{31} a^{15} + \frac{13}{31} a^{14} + \frac{7}{31} a^{13} + \frac{7}{31} a^{11} - \frac{2}{31} a^{10} - \frac{6}{31} a^{9} + \frac{1}{31} a^{8} + \frac{1}{31} a^{7} - \frac{13}{31} a^{6} - \frac{13}{31} a^{5} - \frac{14}{31} a^{4} - \frac{10}{31} a^{3} + \frac{12}{31} a^{2} + \frac{7}{31} a + \frac{5}{31}$, $\frac{1}{5053} a^{19} - \frac{54}{5053} a^{18} - \frac{696}{5053} a^{17} + \frac{537}{5053} a^{16} + \frac{1283}{5053} a^{15} - \frac{73}{163} a^{14} - \frac{1580}{5053} a^{13} + \frac{906}{5053} a^{12} - \frac{2357}{5053} a^{11} + \frac{2456}{5053} a^{10} + \frac{1776}{5053} a^{9} + \frac{2149}{5053} a^{8} + \frac{1732}{5053} a^{7} + \frac{1079}{5053} a^{6} + \frac{2101}{5053} a^{5} - \frac{1128}{5053} a^{4} - \frac{2155}{5053} a^{3} + \frac{2378}{5053} a^{2} + \frac{750}{5053} a - \frac{2497}{5053}$

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:  \( -\frac{92}{31} a^{19} + \frac{627}{31} a^{18} - \frac{2307}{31} a^{17} + \frac{5626}{31} a^{16} - \frac{10118}{31} a^{15} + \frac{13131}{31} a^{14} - \frac{10639}{31} a^{13} - \frac{551}{31} a^{12} + \frac{18282}{31} a^{11} - \frac{35176}{31} a^{10} + \frac{40966}{31} a^{9} - \frac{30588}{31} a^{8} + \frac{7714}{31} a^{7} + \frac{15910}{31} a^{6} - \frac{29382}{31} a^{5} + \frac{29235}{31} a^{4} - \frac{20032}{31} a^{3} + \frac{9527}{31} a^{2} - \frac{2853}{31} a + \frac{412}{31} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{25329}{5053} a^{19} - \frac{152764}{5053} a^{18} + \frac{503951}{5053} a^{17} - \frac{1093755}{5053} a^{16} + \frac{1753840}{5053} a^{15} - \frac{1919087}{5053} a^{14} + \frac{1001249}{5053} a^{13} + \frac{1225227}{5053} a^{12} - \frac{3919571}{5053} a^{11} + \frac{5806927}{5053} a^{10} - \frac{5478540}{5053} a^{9} + \frac{2903809}{5053} a^{8} + \frac{699278}{5053} a^{7} - \frac{3477498}{5053} a^{6} + \frac{4379636}{5053} a^{5} - \frac{3575551}{5053} a^{4} + \frac{2047421}{5053} a^{3} - \frac{820918}{5053} a^{2} + \frac{210511}{5053} a - \frac{32179}{5053} \),  \( \frac{3022}{5053} a^{19} - \frac{8827}{5053} a^{18} + \frac{1018}{5053} a^{17} + \frac{67794}{5053} a^{16} - \frac{224995}{5053} a^{15} + \frac{463451}{5053} a^{14} - \frac{632115}{5053} a^{13} + \frac{514612}{5053} a^{12} + \frac{61697}{5053} a^{11} - \frac{905801}{5053} a^{10} + \frac{1641544}{5053} a^{9} - \frac{1759553}{5053} a^{8} + \frac{1108574}{5053} a^{7} - \frac{9205}{5053} a^{6} - \frac{932803}{5053} a^{5} + \frac{1297157}{5053} a^{4} - \frac{1092986}{5053} a^{3} + \frac{620350}{5053} a^{2} - \frac{220391}{5053} a + \frac{37315}{5053} \),  \( \frac{661}{163} a^{19} - \frac{124928}{5053} a^{18} + \frac{420978}{5053} a^{17} - \frac{938039}{5053} a^{16} + \frac{1551443}{5053} a^{15} - \frac{1776615}{5053} a^{14} + \frac{1074102}{5053} a^{13} + \frac{27388}{163} a^{12} - \frac{3336671}{5053} a^{11} + \frac{5260634}{5053} a^{10} - \frac{5227263}{5053} a^{9} + \frac{3066739}{5053} a^{8} + \frac{287465}{5053} a^{7} - \frac{3055944}{5053} a^{6} + \frac{4126539}{5053} a^{5} - \frac{3480987}{5053} a^{4} + \frac{2043593}{5053} a^{3} - \frac{811525}{5053} a^{2} + \frac{183007}{5053} a - \frac{13041}{5053} \),  \( \frac{9688}{5053} a^{19} - \frac{62514}{5053} a^{18} + \frac{214318}{5053} a^{17} - \frac{481191}{5053} a^{16} + \frac{786940}{5053} a^{15} - \frac{887816}{5053} a^{14} + \frac{489290}{5053} a^{13} + \frac{520726}{5053} a^{12} - \frac{1788219}{5053} a^{11} + \frac{2670558}{5053} a^{10} - \frac{2541073}{5053} a^{9} + \frac{1315747}{5053} a^{8} + \frac{383446}{5053} a^{7} - \frac{1679390}{5053} a^{6} + \frac{2046980}{5053} a^{5} - \frac{1616689}{5053} a^{4} + \frac{867375}{5053} a^{3} - \frac{297016}{5053} a^{2} + \frac{50968}{5053} a + \frac{1850}{5053} \),  \( \frac{8673}{5053} a^{19} - \frac{57093}{5053} a^{18} + \frac{207144}{5053} a^{17} - \frac{501386}{5053} a^{16} + \frac{901654}{5053} a^{15} - \frac{1168227}{5053} a^{14} + \frac{948893}{5053} a^{13} + \frac{45800}{5053} a^{12} - \frac{1626356}{5053} a^{11} + \frac{3146600}{5053} a^{10} - \frac{3668404}{5053} a^{9} + \frac{2743495}{5053} a^{8} - \frac{22298}{163} a^{7} - \frac{1440257}{5053} a^{6} + \frac{2648464}{5053} a^{5} - \frac{2625162}{5053} a^{4} + \frac{1785093}{5053} a^{3} - \frac{837490}{5053} a^{2} + \frac{247669}{5053} a - \frac{29967}{5053} \),  \( \frac{17}{31} a^{19} - \frac{85}{31} a^{18} + \frac{262}{31} a^{17} - \frac{554}{31} a^{16} + \frac{945}{31} a^{15} - \frac{1137}{31} a^{14} + \frac{919}{31} a^{13} + \frac{26}{31} a^{12} - \frac{1440}{31} a^{11} + \frac{3010}{31} a^{10} - \frac{3636}{31} a^{9} + \frac{3142}{31} a^{8} - \frac{1374}{31} a^{7} - \frac{794}{31} a^{6} + \frac{2413}{31} a^{5} - \frac{2876}{31} a^{4} + \frac{2341}{31} a^{3} - \frac{1379}{31} a^{2} + \frac{508}{31} a - \frac{92}{31} \),  \( \frac{21149}{5053} a^{19} - \frac{145627}{5053} a^{18} + \frac{529272}{5053} a^{17} - \frac{1270271}{5053} a^{16} + \frac{2236243}{5053} a^{15} - \frac{2830296}{5053} a^{14} + \frac{2149387}{5053} a^{13} + \frac{409311}{5053} a^{12} - \frac{4278446}{5053} a^{11} + \frac{7746450}{5053} a^{10} - \frac{8690300}{5053} a^{9} + \frac{6097415}{5053} a^{8} - \frac{1074440}{5053} a^{7} - \frac{3827306}{5053} a^{6} + \frac{6377298}{5053} a^{5} - \frac{6047833}{5053} a^{4} + \frac{3968896}{5053} a^{3} - \frac{1797425}{5053} a^{2} + \frac{522635}{5053} a - \frac{71055}{5053} \),  \( \frac{13699}{5053} a^{19} - \frac{91495}{5053} a^{18} + \frac{322432}{5053} a^{17} - \frac{745878}{5053} a^{16} + \frac{1259517}{5053} a^{15} - \frac{1497511}{5053} a^{14} + \frac{967918}{5053} a^{13} + \frac{582221}{5053} a^{12} - \frac{2687906}{5053} a^{11} + \frac{4339463}{5053} a^{10} - \frac{4436001}{5053} a^{9} + \frac{2654665}{5053} a^{8} + \frac{165996}{5053} a^{7} - \frac{2544375}{5053} a^{6} + \frac{3472263}{5053} a^{5} - \frac{2926411}{5053} a^{4} + \frac{1691525}{5053} a^{3} - \frac{629649}{5053} a^{2} + \frac{127989}{5053} a - \frac{5417}{5053} \),  \( \frac{17220}{5053} a^{19} - \frac{116510}{5053} a^{18} + \frac{420158}{5053} a^{17} - \frac{999529}{5053} a^{16} + \frac{1745970}{5053} a^{15} - \frac{2180086}{5053} a^{14} + \frac{1613571}{5053} a^{13} + \frac{412002}{5053} a^{12} - \frac{3398601}{5053} a^{11} + \frac{6027265}{5053} a^{10} - \frac{6641753}{5053} a^{9} + \frac{4570410}{5053} a^{8} - \frac{669978}{5053} a^{7} - \frac{3029182}{5053} a^{6} + \frac{4913375}{5053} a^{5} - \frac{4596376}{5053} a^{4} + \frac{2993138}{5053} a^{3} - \frac{1351459}{5053} a^{2} + \frac{392525}{5053} a - \frac{53708}{5053} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 182.345344476 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_5\wr C_2$ (as 20T48):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 200
The 14 conjugacy class representatives for $D_5\wr C_2$
Character table for $D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.1413.1, 10.0.1533364992.1 x5

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 25 sibling: 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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
2Data not computed
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
157Data not computed