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Properties

Label 20.20.4291343915...0625.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{10}\cdot 41^{19}$
Root discriminant $76.14$
Ramified primes $5, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4099, 129857, -352368, -656657, 1584530, 1291062, -2320717, -1138974, 1631687, 509336, -630959, -125226, 142122, 17616, -18966, -1407, 1465, 59, -60, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 60*x^18 + 59*x^17 + 1465*x^16 - 1407*x^15 - 18966*x^14 + 17616*x^13 + 142122*x^12 - 125226*x^11 - 630959*x^10 + 509336*x^9 + 1631687*x^8 - 1138974*x^7 - 2320717*x^6 + 1291062*x^5 + 1584530*x^4 - 656657*x^3 - 352368*x^2 + 129857*x - 4099)
 
gp: K = bnfinit(x^20 - x^19 - 60*x^18 + 59*x^17 + 1465*x^16 - 1407*x^15 - 18966*x^14 + 17616*x^13 + 142122*x^12 - 125226*x^11 - 630959*x^10 + 509336*x^9 + 1631687*x^8 - 1138974*x^7 - 2320717*x^6 + 1291062*x^5 + 1584530*x^4 - 656657*x^3 - 352368*x^2 + 129857*x - 4099, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 60 x^{18} + 59 x^{17} + 1465 x^{16} - 1407 x^{15} - 18966 x^{14} + 17616 x^{13} + 142122 x^{12} - 125226 x^{11} - 630959 x^{10} + 509336 x^{9} + 1631687 x^{8} - 1138974 x^{7} - 2320717 x^{6} + 1291062 x^{5} + 1584530 x^{4} - 656657 x^{3} - 352368 x^{2} + 129857 x - 4099 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(42913439156921905845805508304306640625=5^{10}\cdot 41^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.14$
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:  $5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(205=5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{205}(1,·)$, $\chi_{205}(66,·)$, $\chi_{205}(9,·)$, $\chi_{205}(74,·)$, $\chi_{205}(141,·)$, $\chi_{205}(144,·)$, $\chi_{205}(81,·)$, $\chi_{205}(146,·)$, $\chi_{205}(84,·)$, $\chi_{205}(86,·)$, $\chi_{205}(31,·)$, $\chi_{205}(16,·)$, $\chi_{205}(39,·)$, $\chi_{205}(169,·)$, $\chi_{205}(49,·)$, $\chi_{205}(114,·)$, $\chi_{205}(51,·)$, $\chi_{205}(201,·)$, $\chi_{205}(184,·)$, $\chi_{205}(159,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{146} a^{17} + \frac{3}{146} a^{16} + \frac{2}{73} a^{15} + \frac{5}{73} a^{14} + \frac{14}{73} a^{13} + \frac{10}{73} a^{12} - \frac{3}{146} a^{11} + \frac{23}{73} a^{10} + \frac{26}{73} a^{9} - \frac{53}{146} a^{8} - \frac{29}{146} a^{7} - \frac{41}{146} a^{6} - \frac{21}{73} a^{5} - \frac{7}{146} a^{4} + \frac{13}{146} a^{3} - \frac{25}{146} a^{2} - \frac{1}{73} a - \frac{21}{73}$, $\frac{1}{146} a^{18} - \frac{5}{146} a^{16} - \frac{1}{73} a^{15} - \frac{1}{73} a^{14} - \frac{32}{73} a^{13} - \frac{63}{146} a^{12} + \frac{55}{146} a^{11} + \frac{30}{73} a^{10} - \frac{63}{146} a^{9} - \frac{8}{73} a^{8} + \frac{23}{73} a^{7} - \frac{65}{146} a^{6} - \frac{27}{146} a^{5} + \frac{17}{73} a^{4} - \frac{32}{73} a^{3} - \frac{1}{2} a^{2} - \frac{18}{73} a - \frac{10}{73}$, $\frac{1}{1488964253466800176022694055635917724026774} a^{19} + \frac{882832655142142848940726436917402343755}{1488964253466800176022694055635917724026774} a^{18} - \frac{2430027096676466838346054663438259950475}{1488964253466800176022694055635917724026774} a^{17} + \frac{6174420776748211914229476345557635815557}{1488964253466800176022694055635917724026774} a^{16} - \frac{16298950222289947961947157338680917670114}{744482126733400088011347027817958862013387} a^{15} + \frac{317449743589993067611314756891277110428793}{744482126733400088011347027817958862013387} a^{14} + \frac{583119269448978213759612970849019625437903}{1488964253466800176022694055635917724026774} a^{13} + \frac{120175109252251468181692305845291045205074}{744482126733400088011347027817958862013387} a^{12} - \frac{531089740490527889477851113094644890147393}{1488964253466800176022694055635917724026774} a^{11} - \frac{95937908312732679744959824232565848948889}{1488964253466800176022694055635917724026774} a^{10} + \frac{739874537223038811130241358937209675066929}{1488964253466800176022694055635917724026774} a^{9} + \frac{219966358570014607122480457003670757956963}{744482126733400088011347027817958862013387} a^{8} - \frac{311012961310511433443899555196077895569759}{1488964253466800176022694055635917724026774} a^{7} - \frac{245144619677826987895565026450969923601649}{744482126733400088011347027817958862013387} a^{6} + \frac{220551329788977977758728465702190491062409}{1488964253466800176022694055635917724026774} a^{5} + \frac{29539479395650201230121020925559079666348}{744482126733400088011347027817958862013387} a^{4} + \frac{393456677060567756996407586464352020001181}{1488964253466800176022694055635917724026774} a^{3} - \frac{536187438310334414760665622019775240360757}{1488964253466800176022694055635917724026774} a^{2} - \frac{11518971683112395942581294486805459479503}{744482126733400088011347027817958862013387} a - \frac{255027005387814829890443481073541108076888}{744482126733400088011347027817958862013387}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $19$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1163330087490 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.1723025.1, 5.5.2825761.1, 10.10.327381934393961.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R $20$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
41Data not computed